% -*- TeX:EN -*- %%% ==================================================================== %%% $RCSfile: Module2Latex.java,v $ %%% $Revision: 1.28 $ %%% ==================================================================== %%% %%% Generated by class com.meyling.principia.latex.Module2Latex %%% at 2004-09-26T23:33:10+0200. %%% %%% Project Hilbert II - http://www.qedeq.org %%% Copyright 2001, 2002, 2003, 2004 Michael Meyling (mime@qedeq.org) %%% %%% This file is part of *Principia Mathematica II*, the prototype of %%% Hilbert II. %%% %%% Permission is granted to copy, distribute and/or modify this document %%% under the terms of the GNU Free Documentation License, Version 1.2 %%% or any later version published by the Free Software Foundation; %%% with no Invariant Sections, no Front-Cover Texts, and no %%% Back-Cover Texts. %%% \documentclass[a4paper]{article} \usepackage{amsmath,amsthm,amsfonts} \usepackage[]{color} \usepackage{xr} \usepackage{epsfig,longtable} \usepackage{varioref} \usepackage[bookmarksnumbered,colorlinks,breaklinks,plainpages,backref,bookmarksopen=true,pdfpagemode=None]{hyperref} \oddsidemargin 8mm \evensidemargin 9mm \topmargin 0mm \headsep 10mm \marginparsep 2.5mm \marginparwidth 25mm \textwidth 160mm \textheight 220mm \newtheorem{axm}{Axiom}[section] \newtheorem{abr}{Abbreviation}[section] \newtheorem{thm}{Theorem}[section] \newtheorem{dec}{Rule Declaration}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}{Proposition} \newtheorem{lem}[thm]{Lemma} \theoremstyle{remark} \newtheorem*{rmk}{Remark} \makeindex \makeatletter \externaldocument{prophilbert3_1.00.00_1.00.00.qedeq} \hyperbaseurl{prophilbert3_1.00.00_1.00.00.qedeq} \makeatother \setlength{\LTleft}{0pt} \setlength{\LTright}{0pt} \setlength{\parindent}{0pt} \frenchspacing \sloppy \pagestyle{headings} \title{Further Theorems of Propositional Calculus} \author{Michael Meyling} \date{\tt } \begin{document} \maketitle \setlongtables {\small This document is part of the project ``Hilbert II''. To get more information about this project look at:\\ \mbox{\url{http://www.qedeq.org}}. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. See also under \url{http://www.gnu.org/copyleft/} } \section*{Abstract} This module includes proofs of popositional calculus theorems. The following theorems and proofs are adapted from D. Hilbert and W. Ackermann's `Grundzuege der theoretischen Logik' (Berlin 1928, Springer) \section*{Specification} This document has the following specification: \begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}} Name: & prophilbert3 \\ Version: & 1.00.00 \\ Rule version: & 1.00.00 \\ Orgin: & \url{http://www.qedeq.org/0_00_53/prophilbert3_1.00.00_1.00.00.qedeq} \\ \end{longtable} \medskip Author of this module: \begin{longtable}[h!]{l@{\extracolsep{\fill}}l} Michael Meyling & mime@qedeq.org \\ \end{longtable} \section*{References} This document uses the results of the following documents: \begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}} Name: & prophilbert2 \\ Version: & 1.00.00 \\ Rule version: & 1.00.00 \\ Orgin: & \url{prophilbert2_1.00.00_1.00.00.qedeq} \\ pdf: & \url{prophilbert2_1.00.00_1.00.00.pdf} \\ \end{longtable} \section*{Content} First distributive law (first direction): \begin{thm}[hilb36] \hypertarget{hilb36}{} \begin{displaymath} (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\end{displaymath} \end{thm} \begin{proof} \mbox{}\\ \begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}} \label{hilb36:1} $1$ & $(P\ \wedge \ Q)\ \rightarrow \ P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb24}{hilb24} } \\ \label{hilb36:2} $2$ & $(P\ \wedge \ A)\ \rightarrow \ P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb36:1]{$1$} } \\ \label{hilb36:3} $3$ & $(B\ \wedge \ A)\ \rightarrow \ B$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb36:2]{$2$} } \\ \label{hilb36:4} $4$ & $(Q\ \wedge \ A)\ \rightarrow \ Q$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb36:3]{$3$} } \\ \label{hilb36:5} $5$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \vee \ P)\ \rightarrow \ (A\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\ \label{hilb36:6} $6$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb36:5]{$5$} } \\ \label{hilb36:7} $7$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb36:6]{$6$} } \\ \label{hilb36:8} $8$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ D)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb36:7]{$7$} } \\ \label{hilb36:9} $9$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P$ in \hyperref[hilb36:8]{$8$} } \\ \label{hilb36:10} $10$ & $(D\ \rightarrow \ Q)\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q$ in \hyperref[hilb36:9]{$9$} } \\ \label{hilb36:11} $11$ & $((Q\ \wedge \ A)\ \rightarrow \ Q)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \wedge \ A$ in \hyperref[hilb36:10]{$10$} } \\ \label{hilb36:12} $12$ & $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:4]{$4$}, \hyperref[hilb36:11]{$11$}} \\ \label{hilb36:13} $13$ & $(P\ \wedge \ Q)\ \rightarrow \ Q$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb25}{hilb25} } \\ \label{hilb36:14} $14$ & $(P\ \wedge \ A)\ \rightarrow \ A$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb36:13]{$13$} } \\ \label{hilb36:15} $15$ & $(B\ \wedge \ A)\ \rightarrow \ A$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb36:14]{$14$} } \\ \label{hilb36:16} $16$ & $(Q\ \wedge \ A)\ \rightarrow \ A$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb36:15]{$15$} } \\ \label{hilb36:17} $17$ & $(D\ \rightarrow \ A)\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $A$ in \hyperref[hilb36:9]{$9$} } \\ \label{hilb36:18} $18$ & $((Q\ \wedge \ A)\ \rightarrow \ A)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \wedge \ A$ in \hyperref[hilb36:17]{$17$} } \\ \label{hilb36:19} $19$ & $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:16]{$16$}, \hyperref[hilb36:18]{$18$}} \\ \label{hilb36:20} $20$ & $P\ \rightarrow \ (Q\ \rightarrow \ (P\ \wedge \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb28}{hilb28} } \\ \label{hilb36:21} $21$ & $P\ \rightarrow \ (A\ \rightarrow \ (P\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb36:20]{$20$} } \\ \label{hilb36:22} $22$ & $B\ \rightarrow \ (A\ \rightarrow \ (B\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb36:21]{$21$} } \\ \label{hilb36:23} $23$ & $B\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (B\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ A$ in \hyperref[hilb36:22]{$22$} } \\ \label{hilb36:24} $24$ & $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb36:23]{$23$} } \\ \label{hilb36:25} $25$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \rightarrow \ P)\ \rightarrow \ (A\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\ \label{hilb36:26} $26$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb36:25]{$25$} } \\ \label{hilb36:27} $27$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb36:26]{$26$} } \\ \label{hilb36:28} $28$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ D)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb36:27]{$27$} } \\ \label{hilb36:29} $29$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb36:28]{$28$} } \\ \label{hilb36:30} $30$ & $(D\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb36:29]{$29$} } \\ \label{hilb36:31} $31$ & $((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \vee \ Q$ in \hyperref[hilb36:30]{$30$} } \\ \label{hilb36:32} $32$ & $((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:24]{$24$}, \hyperref[hilb36:31]{$31$}} \\ \label{hilb36:33} $33$ & $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:12]{$12$}, \hyperref[hilb36:32]{$32$}} \\ \label{hilb36:34} $34$ & $(P\ \rightarrow \ (Q\ \rightarrow \ A))\ \rightarrow \ (Q\ \rightarrow \ (P\ \rightarrow \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb16}{hilb16} } \\ \label{hilb36:35} $35$ & $(P\ \rightarrow \ (Q\ \rightarrow \ B))\ \rightarrow \ (Q\ \rightarrow \ (P\ \rightarrow \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb36:34]{$34$} } \\ \label{hilb36:36} $36$ & $(P\ \rightarrow \ (C\ \rightarrow \ B))\ \rightarrow \ (C\ \rightarrow \ (P\ \rightarrow \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb36:35]{$35$} } \\ \label{hilb36:37} $37$ & $(D\ \rightarrow \ (C\ \rightarrow \ B))\ \rightarrow \ (C\ \rightarrow \ (D\ \rightarrow \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb36:36]{$36$} } \\ \label{hilb36:38} $38$ & $(D\ \rightarrow \ (C\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (C\ \rightarrow \ (D\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb36:37]{$37$} } \\ \label{hilb36:39} $39$ & $(D\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (D\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb36:38]{$38$} } \\ \label{hilb36:40} $40$ & $((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb36:39]{$39$} } \\ \label{hilb36:41} $41$ & $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:33]{$33$}, \hyperref[hilb36:40]{$40$}} \\ \label{hilb36:42} $42$ & $(D\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb36:29]{$29$} } \\ \label{hilb36:43} $43$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \vee \ A$ in \hyperref[hilb36:42]{$42$} } \\ \label{hilb36:44} $44$ & $((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:41]{$41$}, \hyperref[hilb36:43]{$43$}} \\ \label{hilb36:45} $45$ & $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:19]{$19$}, \hyperref[hilb36:44]{$44$}} \\ \label{hilb36:46} $46$ & $(P\ \rightarrow \ (P\ \rightarrow \ Q))\ \rightarrow \ (P\ \rightarrow \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb33}{hilb33} } \\ \label{hilb36:47} $47$ & $(P\ \rightarrow \ (P\ \rightarrow \ A))\ \rightarrow \ (P\ \rightarrow \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb36:46]{$46$} } \\ \label{hilb36:48} $48$ & $(B\ \rightarrow \ (B\ \rightarrow \ A))\ \rightarrow \ (B\ \rightarrow \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb36:47]{$47$} } \\ \label{hilb36:49} $49$ & $(B\ \rightarrow \ (B\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (B\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb36:48]{$48$} } \\ \label{hilb36:50} $50$ & $((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb36:49]{$49$} } \\ \label{hilb36:51} $51$ & $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:45]{$45$}, \hyperref[hilb36:50]{$50$}} \\ & & \qedhere \end{longtable} \end{proof} First distributive law (second direction): \begin{thm}[hilb37] \hypertarget{hilb37}{} \begin{displaymath} ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))\end{displaymath} \end{thm} \begin{proof} \mbox{}\\ \begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}} \label{hilb37:1} $1$ & $P\ \rightarrow \ (Q\ \rightarrow \ (P\ \wedge \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb28}{hilb28} } \\ \label{hilb37:2} $2$ & $P\ \rightarrow \ (A\ \rightarrow \ (P\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb37:1]{$1$} } \\ \label{hilb37:3} $3$ & $B\ \rightarrow \ (A\ \rightarrow \ (B\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb37:2]{$2$} } \\ \label{hilb37:4} $4$ & $Q\ \rightarrow \ (A\ \rightarrow \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb37:3]{$3$} } \\ \label{hilb37:5} $5$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \vee \ P)\ \rightarrow \ (A\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\ \label{hilb37:6} $6$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb37:5]{$5$} } \\ \label{hilb37:7} $7$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb37:6]{$6$} } \\ \label{hilb37:8} $8$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ D)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb37:7]{$7$} } \\ \label{hilb37:9} $9$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P$ in \hyperref[hilb37:8]{$8$} } \\ \label{hilb37:10} $10$ & $(D\ \rightarrow \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q\ \wedge \ A$ in \hyperref[hilb37:9]{$9$} } \\ \label{hilb37:11} $11$ & $(A\ \rightarrow \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $A$ in \hyperref[hilb37:10]{$10$} } \\ \label{hilb37:12} $12$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \rightarrow \ P)\ \rightarrow \ (A\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\ \label{hilb37:13} $13$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb37:12]{$12$} } \\ \label{hilb37:14} $14$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb37:13]{$13$} } \\ \label{hilb37:15} $15$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ D)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb37:14]{$14$} } \\ \label{hilb37:16} $16$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((Q\ \rightarrow \ D)\ \rightarrow \ (Q\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:17} $17$ & $(D\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((Q\ \rightarrow \ D)\ \rightarrow \ (Q\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:16]{$16$} } \\ \label{hilb37:18} $18$ & $((A\ \rightarrow \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((Q\ \rightarrow \ (A\ \rightarrow \ (Q\ \wedge \ A)))\ \rightarrow \ (Q\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $A\ \rightarrow \ (Q\ \wedge \ A)$ in \hyperref[hilb37:17]{$17$} } \\ \label{hilb37:19} $19$ & $(Q\ \rightarrow \ (A\ \rightarrow \ (Q\ \wedge \ A)))\ \rightarrow \ (Q\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:11]{$11$}, \hyperref[hilb37:18]{$18$}} \\ \label{hilb37:20} $20$ & $Q\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:4]{$4$}, \hyperref[hilb37:19]{$19$}} \\ \label{hilb37:21} $21$ & $(P\ \rightarrow \ (Q\ \rightarrow \ A))\ \rightarrow \ (Q\ \rightarrow \ (P\ \rightarrow \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb16}{hilb16} } \\ \label{hilb37:22} $22$ & $(P\ \rightarrow \ (Q\ \rightarrow \ B))\ \rightarrow \ (Q\ \rightarrow \ (P\ \rightarrow \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb37:21]{$21$} } \\ \label{hilb37:23} $23$ & $(P\ \rightarrow \ (C\ \rightarrow \ B))\ \rightarrow \ (C\ \rightarrow \ (P\ \rightarrow \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb37:22]{$22$} } \\ \label{hilb37:24} $24$ & $(D\ \rightarrow \ (C\ \rightarrow \ B))\ \rightarrow \ (C\ \rightarrow \ (D\ \rightarrow \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb37:23]{$23$} } \\ \label{hilb37:25} $25$ & $(D\ \rightarrow \ (C\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (C\ \rightarrow \ (D\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:24]{$24$} } \\ \label{hilb37:26} $26$ & $(D\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (D\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb37:25]{$25$} } \\ \label{hilb37:27} $27$ & $(Q\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (Q\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q$ in \hyperref[hilb37:26]{$26$} } \\ \label{hilb37:28} $28$ & $(P\ \vee \ A)\ \rightarrow \ (Q\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:20]{$20$}, \hyperref[hilb37:27]{$27$}} \\ \label{hilb37:29} $29$ & $(D\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:9]{$9$} } \\ \label{hilb37:30} $30$ & $(Q\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q$ in \hyperref[hilb37:29]{$29$} } \\ \label{hilb37:31} $31$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ D)\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:32} $32$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ D)\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:31]{$31$} } \\ \label{hilb37:33} $33$ & $((Q\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ (Q\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:32]{$32$} } \\ \label{hilb37:34} $34$ & $((P\ \vee \ A)\ \rightarrow \ (Q\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:30]{$30$}, \hyperref[hilb37:33]{$33$}} \\ \label{hilb37:35} $35$ & $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:28]{$28$}, \hyperref[hilb37:34]{$34$}} \\ \label{hilb37:36} $36$ & $(P\ \vee \ (Q\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb14}{hilb14} } \\ \label{hilb37:37} $37$ & $(P\ \vee \ (Q\ \vee \ B))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb37:36]{$36$} } \\ \label{hilb37:38} $38$ & $(P\ \vee \ (C\ \vee \ B))\ \rightarrow \ ((P\ \vee \ C)\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb37:37]{$37$} } \\ \label{hilb37:39} $39$ & $(D\ \vee \ (C\ \vee \ B))\ \rightarrow \ ((D\ \vee \ C)\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb37:38]{$38$} } \\ \label{hilb37:40} $40$ & $(D\ \vee \ (C\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((D\ \vee \ C)\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \wedge \ A$ in \hyperref[hilb37:39]{$39$} } \\ \label{hilb37:41} $41$ & $(D\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((D\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P$ in \hyperref[hilb37:40]{$40$} } \\ \label{hilb37:42} $42$ & $(P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P$ in \hyperref[hilb37:41]{$41$} } \\ \label{hilb37:43} $43$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\ \label{hilb37:44} $44$ & $(\neg P\ \vee \ Q)\ \rightarrow \ (P\ \rightarrow \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\ \label{hilb37:45} $45$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb37:8]{$8$} } \\ \label{hilb37:46} $46$ & $(D\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:45]{$45$} } \\ \label{hilb37:47} $47$ & $((P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:46]{$46$} } \\ \label{hilb37:48} $48$ & $(\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:42]{$42$}, \hyperref[hilb37:47]{$47$}} \\ \label{hilb37:49} $49$ & $(P\ \rightarrow \ B)\ \rightarrow \ (\neg P\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb37:43]{$43$} } \\ \label{hilb37:50} $50$ & $(C\ \rightarrow \ B)\ \rightarrow \ (\neg C\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb37:49]{$49$} } \\ \label{hilb37:51} $51$ & $(C\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg C\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:50]{$50$} } \\ \label{hilb37:52} $52$ & $((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb37:51]{$51$} } \\ \label{hilb37:53} $53$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:54} $54$ & $(D\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:53]{$53$} } \\ \label{hilb37:55} $55$ & $((\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:54]{$54$} } \\ \label{hilb37:56} $56$ & $(((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:48]{$48$}, \hyperref[hilb37:55]{$55$}} \\ \label{hilb37:57} $57$ & $((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:52]{$52$}, \hyperref[hilb37:56]{$56$}} \\ \label{hilb37:58} $58$ & $(\neg P\ \vee \ B)\ \rightarrow \ (P\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb37:44]{$44$} } \\ \label{hilb37:59} $59$ & $(\neg C\ \vee \ B)\ \rightarrow \ (C\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb37:58]{$58$} } \\ \label{hilb37:60} $60$ & $(\neg C\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (C\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:59]{$59$} } \\ \label{hilb37:61} $61$ & $(\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb37:60]{$60$} } \\ \label{hilb37:62} $62$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:53]{$53$} } \\ \label{hilb37:63} $63$ & $((\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:62]{$62$} } \\ \label{hilb37:64} $64$ & $(((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:61]{$61$}, \hyperref[hilb37:63]{$63$}} \\ \label{hilb37:65} $65$ & $((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:57]{$57$}, \hyperref[hilb37:64]{$64$}} \\ \label{hilb37:66} $66$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb37:8]{$8$} } \\ \label{hilb37:67} $67$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:66]{$66$} } \\ \label{hilb37:68} $68$ & $(((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:67]{$67$} } \\ \label{hilb37:69} $69$ & $(\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:65]{$65$}, \hyperref[hilb37:68]{$68$}} \\ \label{hilb37:70} $70$ & $(C\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg C\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:50]{$50$} } \\ \label{hilb37:71} $71$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb37:70]{$70$} } \\ \label{hilb37:72} $72$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:73} $73$ & $(D\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:72]{$72$} } \\ \label{hilb37:74} $74$ & $((\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ in \hyperref[hilb37:73]{$73$} } \\ \label{hilb37:75} $75$ & $(((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:69]{$69$}, \hyperref[hilb37:74]{$74$}} \\ \label{hilb37:76} $76$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:71]{$71$}, \hyperref[hilb37:75]{$75$}} \\ \label{hilb37:77} $77$ & $(\neg C\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (C\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:59]{$59$} } \\ \label{hilb37:78} $78$ & $(\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb37:77]{$77$} } \\ \label{hilb37:79} $79$ & $(D\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:72]{$72$} } \\ \label{hilb37:80} $80$ & $((\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:79]{$79$} } \\ \label{hilb37:81} $81$ & $(((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:78]{$78$}, \hyperref[hilb37:80]{$80$}} \\ \label{hilb37:82} $82$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:76]{$76$}, \hyperref[hilb37:81]{$81$}} \\ \label{hilb37:83} $83$ & $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:35]{$35$}, \hyperref[hilb37:82]{$82$}} \\ \label{hilb37:84} $84$ & $(P\ \vee \ P)\ \rightarrow \ P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb11}{hilb11} } \\ \label{hilb37:85} $85$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((Q\ \wedge \ A)\ \vee \ D)\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \wedge \ A$ in \hyperref[hilb37:8]{$8$} } \\ \label{hilb37:86} $86$ & $(D\ \rightarrow \ P)\ \rightarrow \ (((Q\ \wedge \ A)\ \vee \ D)\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ P))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P$ in \hyperref[hilb37:85]{$85$} } \\ \label{hilb37:87} $87$ & $((P\ \vee \ P)\ \rightarrow \ P)\ \rightarrow \ (((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ P))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \vee \ P$ in \hyperref[hilb37:86]{$86$} } \\ \label{hilb37:88} $88$ & $((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:84]{$84$}, \hyperref[hilb37:87]{$87$}} \\ \label{hilb37:89} $89$ & $(P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\ \label{hilb37:90} $90$ & $(P\ \vee \ B)\ \rightarrow \ (B\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb37:89]{$89$} } \\ \label{hilb37:91} $91$ & $(C\ \vee \ B)\ \rightarrow \ (B\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb37:90]{$90$} } \\ \label{hilb37:92} $92$ & $(C\ \vee \ P)\ \rightarrow \ (P\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P$ in \hyperref[hilb37:91]{$91$} } \\ \label{hilb37:93} $93$ & $((Q\ \wedge \ A)\ \vee \ P)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q\ \wedge \ A$ in \hyperref[hilb37:92]{$92$} } \\ \label{hilb37:94} $94$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ D)\ \rightarrow \ (((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(Q\ \wedge \ A)\ \vee \ (P\ \vee \ P)$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:95} $95$ & $(D\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ D)\ \rightarrow \ (((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:94]{$94$} } \\ \label{hilb37:96} $96$ & $(((Q\ \wedge \ A)\ \vee \ P)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ P))\ \rightarrow \ (((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(Q\ \wedge \ A)\ \vee \ P$ in \hyperref[hilb37:95]{$95$} } \\ \label{hilb37:97} $97$ & $(((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ P))\ \rightarrow \ (((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:93]{$93$}, \hyperref[hilb37:96]{$96$}} \\ \label{hilb37:98} $98$ & $((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:88]{$88$}, \hyperref[hilb37:97]{$97$}} \\ \label{hilb37:99} $99$ & $(C\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \wedge \ A$ in \hyperref[hilb37:91]{$91$} } \\ \label{hilb37:100} $100$ & $((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ P$ in \hyperref[hilb37:99]{$99$} } \\ \label{hilb37:101} $101$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:102} $102$ & $(D\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:101]{$101$} } \\ \label{hilb37:103} $103$ & $(((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P)))\ \rightarrow \ (((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(Q\ \wedge \ A)\ \vee \ (P\ \vee \ P)$ in \hyperref[hilb37:102]{$102$} } \\ \label{hilb37:104} $104$ & $(((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((Q\ \wedge \ A)\ \vee \ (P\ \vee \ P)))\ \rightarrow \ (((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:98]{$98$}, \hyperref[hilb37:103]{$103$}} \\ \label{hilb37:105} $105$ & $((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:100]{$100$}, \hyperref[hilb37:104]{$104$}} \\ \label{hilb37:106} $106$ & $(D\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:45]{$45$} } \\ \label{hilb37:107} $107$ & $(((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:106]{$106$} } \\ \label{hilb37:108} $108$ & $(\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:105]{$105$}, \hyperref[hilb37:107]{$107$}} \\ \label{hilb37:109} $109$ & $(C\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg C\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:50]{$50$} } \\ \label{hilb37:110} $110$ & $((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb37:109]{$109$} } \\ \label{hilb37:111} $111$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:112} $112$ & $(D\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:111]{$111$} } \\ \label{hilb37:113} $113$ & $((\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:112]{$112$} } \\ \label{hilb37:114} $114$ & $(((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:108]{$108$}, \hyperref[hilb37:113]{$113$}} \\ \label{hilb37:115} $115$ & $((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:110]{$110$}, \hyperref[hilb37:114]{$114$}} \\ \label{hilb37:116} $116$ & $(\neg C\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (C\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:59]{$59$} } \\ \label{hilb37:117} $117$ & $(\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb37:116]{$116$} } \\ \label{hilb37:118} $118$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:111]{$111$} } \\ \label{hilb37:119} $119$ & $((\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:118]{$118$} } \\ \label{hilb37:120} $120$ & $(((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:117]{$117$}, \hyperref[hilb37:119]{$119$}} \\ \label{hilb37:121} $121$ & $((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:115]{$115$}, \hyperref[hilb37:120]{$120$}} \\ \label{hilb37:122} $122$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:66]{$66$} } \\ \label{hilb37:123} $123$ & $(((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:122]{$122$} } \\ \label{hilb37:124} $124$ & $(\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:121]{$121$}, \hyperref[hilb37:123]{$123$}} \\ \label{hilb37:125} $125$ & $(C\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg C\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:50]{$50$} } \\ \label{hilb37:126} $126$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb37:125]{$125$} } \\ \label{hilb37:127} $127$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:15]{$15$} } \\ \label{hilb37:128} $128$ & $(D\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:127]{$127$} } \\ \label{hilb37:129} $129$ & $((\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:128]{$128$} } \\ \label{hilb37:130} $130$ & $(((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:124]{$124$}, \hyperref[hilb37:129]{$129$}} \\ \label{hilb37:131} $131$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:126]{$126$}, \hyperref[hilb37:130]{$130$}} \\ \label{hilb37:132} $132$ & $(\neg C\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (C\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb37:59]{$59$} } \\ \label{hilb37:133} $133$ & $(\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb37:132]{$132$} } \\ \label{hilb37:134} $134$ & $(D\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:127]{$127$} } \\ \label{hilb37:135} $135$ & $((\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ ((((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ in \hyperref[hilb37:134]{$134$} } \\ \label{hilb37:136} $136$ & $(((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))\ \rightarrow \ (((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:133]{$133$}, \hyperref[hilb37:135]{$135$}} \\ \label{hilb37:137} $137$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ P)\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:131]{$131$}, \hyperref[hilb37:136]{$136$}} \\ \label{hilb37:138} $138$ & $(P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:83]{$83$}, \hyperref[hilb37:137]{$137$}} \\ \label{hilb37:139} $139$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (D\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb37:25]{$25$} } \\ \label{hilb37:140} $140$ & $((P\ \vee \ A)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \vee \ A$ in \hyperref[hilb37:139]{$139$} } \\ \label{hilb37:141} $141$ & $(P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:138]{$138$}, \hyperref[hilb37:140]{$140$}} \\ \label{hilb37:142} $142$ & $(P\ \rightarrow \ (Q\ \rightarrow \ A))\ \rightarrow \ ((P\ \wedge \ Q)\ \rightarrow \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb29}{hilb29} } \\ \label{hilb37:143} $143$ & $(P\ \rightarrow \ (Q\ \rightarrow \ B))\ \rightarrow \ ((P\ \wedge \ Q)\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb37:142]{$142$} } \\ \label{hilb37:144} $144$ & $(P\ \rightarrow \ (C\ \rightarrow \ B))\ \rightarrow \ ((P\ \wedge \ C)\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb37:143]{$143$} } \\ \label{hilb37:145} $145$ & $(D\ \rightarrow \ (C\ \rightarrow \ B))\ \rightarrow \ ((D\ \wedge \ C)\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb37:144]{$144$} } \\ \label{hilb37:146} $146$ & $(D\ \rightarrow \ (C\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((D\ \wedge \ C)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb37:145]{$145$} } \\ \label{hilb37:147} $147$ & $(D\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((D\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb37:146]{$146$} } \\ \label{hilb37:148} $148$ & $((P\ \vee \ Q)\ \rightarrow \ ((P\ \vee \ A)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \vee \ Q$ in \hyperref[hilb37:147]{$147$} } \\ \label{hilb37:149} $149$ & $((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:141]{$141$}, \hyperref[hilb37:148]{$148$}} \\ & & \qedhere \end{longtable} \end{proof} A form for the abbreviation rule form for disjunction (first direction): \begin{thm}[hilb38] \hypertarget{hilb38}{} \begin{displaymath} (P\ \vee \ Q)\ \rightarrow \ \neg (\neg P\ \wedge \ \neg Q)\end{displaymath} \end{thm} \begin{proof} \mbox{}\\ \begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}} \label{hilb38:1} $1$ & $P\ \rightarrow \ \neg \neg P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb5}{hilb5} } \\ \label{hilb38:2} $2$ & $A\ \rightarrow \ \neg \neg A$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $A$ in \hyperref[hilb38:1]{$1$} } \\ \label{hilb38:3} $3$ & $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb38:2]{$2$} } \\ \label{hilb38:4} $4$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\ \label{hilb38:5} $5$ & $(\neg P\ \vee \ Q)\ \rightarrow \ (P\ \rightarrow \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\ \label{hilb38:6} $6$ & $(P\ \rightarrow \ A)\ \rightarrow \ (\neg P\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb38:4]{$4$} } \\ \label{hilb38:7} $7$ & $(B\ \rightarrow \ A)\ \rightarrow \ (\neg B\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb38:6]{$6$} } \\ \label{hilb38:8} $8$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \rightarrow \ P)\ \rightarrow \ (A\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\ \label{hilb38:9} $9$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb38:8]{$8$} } \\ \label{hilb38:10} $10$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb38:9]{$9$} } \\ \label{hilb38:11} $11$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ D)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb38:10]{$10$} } \\ \label{hilb38:12} $12$ & $(\neg P\ \vee \ A)\ \rightarrow \ (P\ \rightarrow \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb38:5]{$5$} } \\ \label{hilb38:13} $13$ & $(\neg B\ \vee \ A)\ \rightarrow \ (B\ \rightarrow \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb38:12]{$12$} } \\ \label{hilb38:14} $14$ & $Q\ \rightarrow \ \neg \neg Q$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $Q$ in \hyperref[hilb38:1]{$1$} } \\ \label{hilb38:15} $15$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \vee \ P)\ \rightarrow \ (A\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\ \label{hilb38:16} $16$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb38:15]{$15$} } \\ \label{hilb38:17} $17$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb38:16]{$16$} } \\ \label{hilb38:18} $18$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ D)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb38:17]{$17$} } \\ \label{hilb38:19} $19$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((Q\ \vee \ D)\ \rightarrow \ (Q\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb38:18]{$18$} } \\ \label{hilb38:20} $20$ & $(D\ \rightarrow \ \neg \neg P)\ \rightarrow \ ((Q\ \vee \ D)\ \rightarrow \ (Q\ \vee \ \neg \neg P))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg P$ in \hyperref[hilb38:19]{$19$} } \\ \label{hilb38:21} $21$ & $(P\ \rightarrow \ \neg \neg P)\ \rightarrow \ ((Q\ \vee \ P)\ \rightarrow \ (Q\ \vee \ \neg \neg P))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P$ in \hyperref[hilb38:20]{$20$} } \\ \label{hilb38:22} $22$ & $(Q\ \vee \ P)\ \rightarrow \ (Q\ \vee \ \neg \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:1]{$1$}, \hyperref[hilb38:21]{$21$}} \\ \label{hilb38:23} $23$ & $(P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\ \label{hilb38:24} $24$ & $(P\ \vee \ A)\ \rightarrow \ (A\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb38:23]{$23$} } \\ \label{hilb38:25} $25$ & $(B\ \vee \ A)\ \rightarrow \ (A\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb38:24]{$24$} } \\ \label{hilb38:26} $26$ & $(B\ \vee \ \neg \neg P)\ \rightarrow \ (\neg \neg P\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg P$ in \hyperref[hilb38:25]{$25$} } \\ \label{hilb38:27} $27$ & $(Q\ \vee \ \neg \neg P)\ \rightarrow \ (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb38:26]{$26$} } \\ \label{hilb38:28} $28$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((Q\ \vee \ P)\ \rightarrow \ D)\ \rightarrow \ ((Q\ \vee \ P)\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \vee \ P$ in \hyperref[hilb38:11]{$11$} } \\ \label{hilb38:29} $29$ & $(D\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (((Q\ \vee \ P)\ \rightarrow \ D)\ \rightarrow \ ((Q\ \vee \ P)\ \rightarrow \ (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg P\ \vee \ Q$ in \hyperref[hilb38:28]{$28$} } \\ \label{hilb38:30} $30$ & $((Q\ \vee \ \neg \neg P)\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (((Q\ \vee \ P)\ \rightarrow \ (Q\ \vee \ \neg \neg P))\ \rightarrow \ ((Q\ \vee \ P)\ \rightarrow \ (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \vee \ \neg \neg P$ in \hyperref[hilb38:29]{$29$} } \\ \label{hilb38:31} $31$ & $((Q\ \vee \ P)\ \rightarrow \ (Q\ \vee \ \neg \neg P))\ \rightarrow \ ((Q\ \vee \ P)\ \rightarrow \ (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:27]{$27$}, \hyperref[hilb38:30]{$30$}} \\ \label{hilb38:32} $32$ & $(Q\ \vee \ P)\ \rightarrow \ (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:22]{$22$}, \hyperref[hilb38:31]{$31$}} \\ \label{hilb38:33} $33$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ D)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb38:11]{$11$} } \\ \label{hilb38:34} $34$ & $(D\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ D)\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg P\ \vee \ Q$ in \hyperref[hilb38:33]{$33$} } \\ \label{hilb38:35} $35$ & $((Q\ \vee \ P)\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ P))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \vee \ P$ in \hyperref[hilb38:34]{$34$} } \\ \label{hilb38:36} $36$ & $((P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ P))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:32]{$32$}, \hyperref[hilb38:35]{$35$}} \\ \label{hilb38:37} $37$ & $(P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:23]{$23$}, \hyperref[hilb38:36]{$36$}} \\ \label{hilb38:38} $38$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg Q\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\ \label{hilb38:39} $39$ & $(P\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb38:38]{$38$} } \\ \label{hilb38:40} $40$ & $(B\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb38:39]{$39$} } \\ \label{hilb38:41} $41$ & $(B\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg P\ \vee \ Q$ in \hyperref[hilb38:40]{$40$} } \\ \label{hilb38:42} $42$ & $((P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb38:41]{$41$} } \\ \label{hilb38:43} $43$ & $\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:37]{$37$}, \hyperref[hilb38:42]{$42$}} \\ \label{hilb38:44} $44$ & $(B\ \rightarrow \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb38:40]{$40$} } \\ \label{hilb38:45} $45$ & $(\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:44]{$44$} } \\ \label{hilb38:46} $46$ & $\neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:43]{$43$}, \hyperref[hilb38:45]{$45$}} \\ \label{hilb38:47} $47$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb38:18]{$18$} } \\ \label{hilb38:48} $48$ & $(D\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:47]{$47$} } \\ \label{hilb38:49} $49$ & $(\neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb38:48]{$48$} } \\ \label{hilb38:50} $50$ & $(\neg (P\ \vee \ Q)\ \vee \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:46]{$46$}, \hyperref[hilb38:49]{$49$}} \\ \label{hilb38:51} $51$ & $(B\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg B\ \vee \ \neg \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb38:7]{$7$} } \\ \label{hilb38:52} $52$ & $((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb38:51]{$51$} } \\ \label{hilb38:53} $53$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb38:11]{$11$} } \\ \label{hilb38:54} $54$ & $(D\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:53]{$53$} } \\ \label{hilb38:55} $55$ & $((\neg (P\ \vee \ Q)\ \vee \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb38:54]{$54$} } \\ \label{hilb38:56} $56$ & $(((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:50]{$50$}, \hyperref[hilb38:55]{$55$}} \\ \label{hilb38:57} $57$ & $((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:52]{$52$}, \hyperref[hilb38:56]{$56$}} \\ \label{hilb38:58} $58$ & $(\neg B\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (B\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:13]{$13$} } \\ \label{hilb38:59} $59$ & $(\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb38:58]{$58$} } \\ \label{hilb38:60} $60$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:53]{$53$} } \\ \label{hilb38:61} $61$ & $((\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:60]{$60$} } \\ \label{hilb38:62} $62$ & $(((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:59]{$59$}, \hyperref[hilb38:61]{$61$}} \\ \label{hilb38:63} $63$ & $((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:57]{$57$}, \hyperref[hilb38:62]{$62$}} \\ \label{hilb38:64} $64$ & $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:3]{$3$}, \hyperref[hilb38:63]{$63$}} \\ \label{hilb38:65} $65$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg \neg P\ \vee \ D)\ \rightarrow \ (\neg \neg P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg P$ in \hyperref[hilb38:18]{$18$} } \\ \label{hilb38:66} $66$ & $(D\ \rightarrow \ \neg \neg Q)\ \rightarrow \ ((\neg \neg P\ \vee \ D)\ \rightarrow \ (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg Q$ in \hyperref[hilb38:65]{$65$} } \\ \label{hilb38:67} $67$ & $(Q\ \rightarrow \ \neg \neg Q)\ \rightarrow \ ((\neg \neg P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q$ in \hyperref[hilb38:66]{$66$} } \\ \label{hilb38:68} $68$ & $(\neg \neg P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ \neg \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:14]{$14$}, \hyperref[hilb38:67]{$67$}} \\ \label{hilb38:69} $69$ & $(B\ \rightarrow \ (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (\neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg P\ \vee \ \neg \neg Q$ in \hyperref[hilb38:40]{$40$} } \\ \label{hilb38:70} $70$ & $((\neg \neg P\ \vee \ Q)\ \rightarrow \ (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (\neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg P\ \vee \ Q$ in \hyperref[hilb38:69]{$69$} } \\ \label{hilb38:71} $71$ & $\neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:68]{$68$}, \hyperref[hilb38:70]{$70$}} \\ \label{hilb38:72} $72$ & $(B\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:40]{$40$} } \\ \label{hilb38:73} $73$ & $(\neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb38:72]{$72$} } \\ \label{hilb38:74} $74$ & $\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:71]{$71$}, \hyperref[hilb38:73]{$73$}} \\ \label{hilb38:75} $75$ & $(D\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb38:47]{$47$} } \\ \label{hilb38:76} $76$ & $(\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:75]{$75$} } \\ \label{hilb38:77} $77$ & $(\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:74]{$74$}, \hyperref[hilb38:76]{$76$}} \\ \label{hilb38:78} $78$ & $(B\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg B\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:7]{$7$} } \\ \label{hilb38:79} $79$ & $((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb38:78]{$78$} } \\ \label{hilb38:80} $80$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:11]{$11$} } \\ \label{hilb38:81} $81$ & $(D\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb38:80]{$80$} } \\ \label{hilb38:82} $82$ & $((\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb38:81]{$81$} } \\ \label{hilb38:83} $83$ & $(((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:77]{$77$}, \hyperref[hilb38:82]{$82$}} \\ \label{hilb38:84} $84$ & $((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:79]{$79$}, \hyperref[hilb38:83]{$83$}} \\ \label{hilb38:85} $85$ & $(\neg B\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (B\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb38:13]{$13$} } \\ \label{hilb38:86} $86$ & $(\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb38:85]{$85$} } \\ \label{hilb38:87} $87$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb38:80]{$80$} } \\ \label{hilb38:88} $88$ & $((\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb38:87]{$87$} } \\ \label{hilb38:89} $89$ & $(((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:86]{$86$}, \hyperref[hilb38:88]{$88$}} \\ \label{hilb38:90} $90$ & $((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:84]{$84$}, \hyperref[hilb38:89]{$89$}} \\ \label{hilb38:91} $91$ & $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:64]{$64$}, \hyperref[hilb38:90]{$90$}} \\ \label{hilb38:92} $92$ & $(P\ \vee \ Q)\ \rightarrow \ \neg (\neg P\ \wedge \ \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb38:91]{$91$} at occurence $1$ } \\ & & \qedhere \end{longtable} \end{proof} A form for the abbreviation rule form for disjunction (second direction): \begin{thm}[hilb39] \hypertarget{hilb39}{} \begin{displaymath} \neg (\neg P\ \wedge \ \neg Q)\ \rightarrow \ (P\ \vee \ Q)\end{displaymath} \end{thm} \begin{proof} \mbox{}\\ \begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}} \label{hilb39:1} $1$ & $\neg \neg P\ \rightarrow \ P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb6}{hilb6} } \\ \label{hilb39:2} $2$ & $\neg \neg A\ \rightarrow \ A$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $A$ in \hyperref[hilb39:1]{$1$} } \\ \label{hilb39:3} $3$ & $\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb39:2]{$2$} } \\ \label{hilb39:4} $4$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\ \label{hilb39:5} $5$ & $(\neg P\ \vee \ Q)\ \rightarrow \ (P\ \rightarrow \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\ \label{hilb39:6} $6$ & $(P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\ \label{hilb39:7} $7$ & $(P\ \vee \ A)\ \rightarrow \ (A\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb39:6]{$6$} } \\ \label{hilb39:8} $8$ & $(B\ \vee \ A)\ \rightarrow \ (A\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb39:7]{$7$} } \\ \label{hilb39:9} $9$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \rightarrow \ P)\ \rightarrow \ (A\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\ \label{hilb39:10} $10$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb39:9]{$9$} } \\ \label{hilb39:11} $11$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb39:10]{$10$} } \\ \label{hilb39:12} $12$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ D)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb39:11]{$11$} } \\ \label{hilb39:13} $13$ & $(B\ \vee \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb39:8]{$8$} } \\ \label{hilb39:14} $14$ & $(P\ \rightarrow \ A)\ \rightarrow \ (\neg P\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb39:4]{$4$} } \\ \label{hilb39:15} $15$ & $(B\ \rightarrow \ A)\ \rightarrow \ (\neg B\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb39:14]{$14$} } \\ \label{hilb39:16} $16$ & $(B\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg B\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb39:15]{$15$} } \\ \label{hilb39:17} $17$ & $(\neg P\ \vee \ A)\ \rightarrow \ (P\ \rightarrow \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb39:5]{$5$} } \\ \label{hilb39:18} $18$ & $(\neg B\ \vee \ A)\ \rightarrow \ (B\ \rightarrow \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb39:17]{$17$} } \\ \label{hilb39:19} $19$ & $(\neg B\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (B\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb39:18]{$18$} } \\ \label{hilb39:20} $20$ & $\neg \neg Q\ \rightarrow \ Q$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $Q$ in \hyperref[hilb39:1]{$1$} } \\ \label{hilb39:21} $21$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \vee \ P)\ \rightarrow \ (A\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\ \label{hilb39:22} $22$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb39:21]{$21$} } \\ \label{hilb39:23} $23$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb39:22]{$22$} } \\ \label{hilb39:24} $24$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ D)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb39:23]{$23$} } \\ \label{hilb39:25} $25$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((Q\ \vee \ D)\ \rightarrow \ (Q\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb39:24]{$24$} } \\ \label{hilb39:26} $26$ & $(D\ \rightarrow \ P)\ \rightarrow \ ((Q\ \vee \ D)\ \rightarrow \ (Q\ \vee \ P))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P$ in \hyperref[hilb39:25]{$25$} } \\ \label{hilb39:27} $27$ & $(\neg \neg P\ \rightarrow \ P)\ \rightarrow \ ((Q\ \vee \ \neg \neg P)\ \rightarrow \ (Q\ \vee \ P))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg P$ in \hyperref[hilb39:26]{$26$} } \\ \label{hilb39:28} $28$ & $(Q\ \vee \ \neg \neg P)\ \rightarrow \ (Q\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:1]{$1$}, \hyperref[hilb39:27]{$27$}} \\ \label{hilb39:29} $29$ & $(B\ \vee \ P)\ \rightarrow \ (P\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P$ in \hyperref[hilb39:8]{$8$} } \\ \label{hilb39:30} $30$ & $(Q\ \vee \ P)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q$ in \hyperref[hilb39:29]{$29$} } \\ \label{hilb39:31} $31$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((Q\ \vee \ \neg \neg P)\ \rightarrow \ D)\ \rightarrow \ ((Q\ \vee \ \neg \neg P)\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \vee \ \neg \neg P$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:32} $32$ & $(D\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (((Q\ \vee \ \neg \neg P)\ \rightarrow \ D)\ \rightarrow \ ((Q\ \vee \ \neg \neg P)\ \rightarrow \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb39:31]{$31$} } \\ \label{hilb39:33} $33$ & $((Q\ \vee \ P)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (((Q\ \vee \ \neg \neg P)\ \rightarrow \ (Q\ \vee \ P))\ \rightarrow \ ((Q\ \vee \ \neg \neg P)\ \rightarrow \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \vee \ P$ in \hyperref[hilb39:32]{$32$} } \\ \label{hilb39:34} $34$ & $((Q\ \vee \ \neg \neg P)\ \rightarrow \ (Q\ \vee \ P))\ \rightarrow \ ((Q\ \vee \ \neg \neg P)\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:30]{$30$}, \hyperref[hilb39:33]{$33$}} \\ \label{hilb39:35} $35$ & $(Q\ \vee \ \neg \neg P)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:28]{$28$}, \hyperref[hilb39:34]{$34$}} \\ \label{hilb39:36} $36$ & $(B\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $Q$ in \hyperref[hilb39:8]{$8$} } \\ \label{hilb39:37} $37$ & $(\neg \neg P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ \neg \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg P$ in \hyperref[hilb39:36]{$36$} } \\ \label{hilb39:38} $38$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg P\ \vee \ Q)\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg P\ \vee \ Q)\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg P\ \vee \ Q$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:39} $39$ & $(D\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (((\neg \neg P\ \vee \ Q)\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb39:38]{$38$} } \\ \label{hilb39:40} $40$ & $((Q\ \vee \ \neg \neg P)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (((\neg \neg P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ \neg \neg P))\ \rightarrow \ ((\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \vee \ \neg \neg P$ in \hyperref[hilb39:39]{$39$} } \\ \label{hilb39:41} $41$ & $((\neg \neg P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ \neg \neg P))\ \rightarrow \ ((\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:35]{$35$}, \hyperref[hilb39:40]{$40$}} \\ \label{hilb39:42} $42$ & $(\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:37]{$37$}, \hyperref[hilb39:41]{$41$}} \\ \label{hilb39:43} $43$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg Q\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\ \label{hilb39:44} $44$ & $(P\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb39:43]{$43$} } \\ \label{hilb39:45} $45$ & $(B\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb39:44]{$44$} } \\ \label{hilb39:46} $46$ & $(B\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb39:45]{$45$} } \\ \label{hilb39:47} $47$ & $((\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg P\ \vee \ Q$ in \hyperref[hilb39:46]{$46$} } \\ \label{hilb39:48} $48$ & $\neg (P\ \vee \ Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:42]{$42$}, \hyperref[hilb39:47]{$47$}} \\ \label{hilb39:49} $49$ & $(B\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:45]{$45$} } \\ \label{hilb39:50} $50$ & $(\neg (P\ \vee \ Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb39:49]{$49$} } \\ \label{hilb39:51} $51$ & $\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:48]{$48$}, \hyperref[hilb39:50]{$50$}} \\ \label{hilb39:52} $52$ & $(B\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb39:45]{$45$} } \\ \label{hilb39:53} $53$ & $(\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:52]{$52$} } \\ \label{hilb39:54} $54$ & $\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:51]{$51$}, \hyperref[hilb39:53]{$53$}} \\ \label{hilb39:55} $55$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb39:24]{$24$} } \\ \label{hilb39:56} $56$ & $(D\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:55]{$55$} } \\ \label{hilb39:57} $57$ & $(\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb39:56]{$56$} } \\ \label{hilb39:58} $58$ & $((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:54]{$54$}, \hyperref[hilb39:57]{$57$}} \\ \label{hilb39:59} $59$ & $(B\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:8]{$8$} } \\ \label{hilb39:60} $60$ & $((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb39:59]{$59$} } \\ \label{hilb39:61} $61$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:62} $62$ & $(D\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:61]{$61$} } \\ \label{hilb39:63} $63$ & $(((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:62]{$62$} } \\ \label{hilb39:64} $64$ & $(((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:60]{$60$}, \hyperref[hilb39:63]{$63$}} \\ \label{hilb39:65} $65$ & $((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:58]{$58$}, \hyperref[hilb39:64]{$64$}} \\ \label{hilb39:66} $66$ & $(\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb39:13]{$13$} } \\ \label{hilb39:67} $67$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:68} $68$ & $(D\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:67]{$67$} } \\ \label{hilb39:69} $69$ & $(((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb39:68]{$68$} } \\ \label{hilb39:70} $70$ & $((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:65]{$65$}, \hyperref[hilb39:69]{$69$}} \\ \label{hilb39:71} $71$ & $(\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:66]{$66$}, \hyperref[hilb39:70]{$70$}} \\ \label{hilb39:72} $72$ & $(\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb39:16]{$16$} } \\ \label{hilb39:73} $73$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:74} $74$ & $(D\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:73]{$73$} } \\ \label{hilb39:75} $75$ & $((\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:74]{$74$} } \\ \label{hilb39:76} $76$ & $((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:71]{$71$}, \hyperref[hilb39:75]{$75$}} \\ \label{hilb39:77} $77$ & $(\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:72]{$72$}, \hyperref[hilb39:76]{$76$}} \\ \label{hilb39:78} $78$ & $(\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:19]{$19$} } \\ \label{hilb39:79} $79$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)$ in \hyperref[hilb39:73]{$73$} } \\ \label{hilb39:80} $80$ & $((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:79]{$79$} } \\ \label{hilb39:81} $81$ & $((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:78]{$78$}, \hyperref[hilb39:80]{$80$}} \\ \label{hilb39:82} $82$ & $(\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:77]{$77$}, \hyperref[hilb39:81]{$81$}} \\ \label{hilb39:83} $83$ & $\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:3]{$3$}, \hyperref[hilb39:82]{$82$}} \\ \label{hilb39:84} $84$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg \neg P\ \vee \ D)\ \rightarrow \ (\neg \neg P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg P$ in \hyperref[hilb39:24]{$24$} } \\ \label{hilb39:85} $85$ & $(D\ \rightarrow \ Q)\ \rightarrow \ ((\neg \neg P\ \vee \ D)\ \rightarrow \ (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q$ in \hyperref[hilb39:84]{$84$} } \\ \label{hilb39:86} $86$ & $(\neg \neg Q\ \rightarrow \ Q)\ \rightarrow \ ((\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg Q$ in \hyperref[hilb39:85]{$85$} } \\ \label{hilb39:87} $87$ & $(\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:20]{$20$}, \hyperref[hilb39:86]{$86$}} \\ \label{hilb39:88} $88$ & $(B\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg P\ \vee \ Q$ in \hyperref[hilb39:45]{$45$} } \\ \label{hilb39:89} $89$ & $((\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg P\ \vee \ \neg \neg Q$ in \hyperref[hilb39:88]{$88$} } \\ \label{hilb39:90} $90$ & $\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ \neg \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:87]{$87$}, \hyperref[hilb39:89]{$89$}} \\ \label{hilb39:91} $91$ & $(B\ \rightarrow \ \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb39:45]{$45$} } \\ \label{hilb39:92} $92$ & $(\neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:91]{$91$} } \\ \label{hilb39:93} $93$ & $\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:90]{$90$}, \hyperref[hilb39:92]{$92$}} \\ \label{hilb39:94} $94$ & $(B\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:45]{$45$} } \\ \label{hilb39:95} $95$ & $(\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb39:94]{$94$} } \\ \label{hilb39:96} $96$ & $\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:93]{$93$}, \hyperref[hilb39:95]{$95$}} \\ \label{hilb39:97} $97$ & $(D\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb39:55]{$55$} } \\ \label{hilb39:98} $98$ & $(\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:97]{$97$} } \\ \label{hilb39:99} $99$ & $((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:96]{$96$}, \hyperref[hilb39:98]{$98$}} \\ \label{hilb39:100} $100$ & $(B\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb39:8]{$8$} } \\ \label{hilb39:101} $101$ & $((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb39:100]{$100$} } \\ \label{hilb39:102} $102$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:103} $103$ & $(D\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:102]{$102$} } \\ \label{hilb39:104} $104$ & $(((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb39:103]{$103$} } \\ \label{hilb39:105} $105$ & $(((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)))\ \rightarrow \ (((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:101]{$101$}, \hyperref[hilb39:104]{$104$}} \\ \label{hilb39:106} $106$ & $((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:99]{$99$}, \hyperref[hilb39:105]{$105$}} \\ \label{hilb39:107} $107$ & $(\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:13]{$13$} } \\ \label{hilb39:108} $108$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:109} $109$ & $(D\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:108]{$108$} } \\ \label{hilb39:110} $110$ & $(((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:109]{$109$} } \\ \label{hilb39:111} $111$ & $((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ ((P\ \vee \ Q)\ \vee \ \neg \neg \neg (\neg \neg P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:106]{$106$}, \hyperref[hilb39:110]{$110$}} \\ \label{hilb39:112} $112$ & $(\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:107]{$107$}, \hyperref[hilb39:111]{$111$}} \\ \label{hilb39:113} $113$ & $(\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg P\ \vee \ Q)$ in \hyperref[hilb39:16]{$16$} } \\ \label{hilb39:114} $114$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)$ in \hyperref[hilb39:12]{$12$} } \\ \label{hilb39:115} $115$ & $(D\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:114]{$114$} } \\ \label{hilb39:116} $116$ & $((\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:115]{$115$} } \\ \label{hilb39:117} $117$ & $((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:112]{$112$}, \hyperref[hilb39:116]{$116$}} \\ \label{hilb39:118} $118$ & $(\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:113]{$113$}, \hyperref[hilb39:117]{$117$}} \\ \label{hilb39:119} $119$ & $(\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)$ in \hyperref[hilb39:19]{$19$} } \\ \label{hilb39:120} $120$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q)$ in \hyperref[hilb39:114]{$114$} } \\ \label{hilb39:121} $121$ & $((\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q)))\ \rightarrow \ (((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)$ in \hyperref[hilb39:120]{$120$} } \\ \label{hilb39:122} $122$ & $((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \vee \ (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:119]{$119$}, \hyperref[hilb39:121]{$121$}} \\ \label{hilb39:123} $123$ & $(\neg \neg (\neg \neg P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:118]{$118$}, \hyperref[hilb39:122]{$122$}} \\ \label{hilb39:124} $124$ & $\neg \neg (\neg \neg P\ \vee \ \neg \neg Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:83]{$83$}, \hyperref[hilb39:123]{$123$}} \\ \label{hilb39:125} $125$ & $\neg (\neg P\ \wedge \ \neg Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb39:124]{$124$} at occurence $1$ } \\ & & \qedhere \end{longtable} \end{proof} By duality we get the second distributive law (first direction): \begin{thm}[hilb40] \hypertarget{hilb40}{} \begin{displaymath} (P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\end{displaymath} \end{thm} \begin{proof} \mbox{}\\ \begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}} \label{hilb40:1} $1$ & $((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb37}{hilb37} } \\ \label{hilb40:2} $2$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg Q\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\ \label{hilb40:3} $3$ & $(P\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb40:2]{$2$} } \\ \label{hilb40:4} $4$ & $(B\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb40:3]{$3$} } \\ \label{hilb40:5} $5$ & $(B\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb40:4]{$4$} } \\ \label{hilb40:6} $6$ & $(((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb40:5]{$5$} } \\ \label{hilb40:7} $7$ & $\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:1]{$1$}, \hyperref[hilb40:6]{$6$}} \\ \label{hilb40:8} $8$ & $P\ \rightarrow \ \neg \neg P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb5}{hilb5} } \\ \label{hilb40:9} $9$ & $B\ \rightarrow \ \neg \neg B$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb40:8]{$8$} } \\ \label{hilb40:10} $10$ & $(Q\ \wedge \ A)\ \rightarrow \ \neg \neg (Q\ \wedge \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \wedge \ A$ in \hyperref[hilb40:9]{$9$} } \\ \label{hilb40:11} $11$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \vee \ P)\ \rightarrow \ (A\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\ \label{hilb40:12} $12$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb40:11]{$11$} } \\ \label{hilb40:13} $13$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb40:12]{$12$} } \\ \label{hilb40:14} $14$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ D)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb40:13]{$13$} } \\ \label{hilb40:15} $15$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:16} $16$ & $(D\ \rightarrow \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (Q\ \wedge \ A)$ in \hyperref[hilb40:15]{$15$} } \\ \label{hilb40:17} $17$ & $((Q\ \wedge \ A)\ \rightarrow \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $Q\ \wedge \ A$ in \hyperref[hilb40:16]{$16$} } \\ \label{hilb40:18} $18$ & $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:10]{$10$}, \hyperref[hilb40:17]{$17$}} \\ \label{hilb40:19} $19$ & $(P\ \rightarrow \ B)\ \rightarrow \ (\neg B\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:2]{$2$} } \\ \label{hilb40:20} $20$ & $(C\ \rightarrow \ B)\ \rightarrow \ (\neg B\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:19]{$19$} } \\ \label{hilb40:21} $21$ & $(C\ \rightarrow \ (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ \neg \neg (Q\ \wedge \ A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:22} $22$ & $((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb40:21]{$21$} } \\ \label{hilb40:23} $23$ & $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:18]{$18$}, \hyperref[hilb40:22]{$22$}} \\ \label{hilb40:24} $24$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\ \label{hilb40:25} $25$ & $(\neg P\ \vee \ Q)\ \rightarrow \ (P\ \rightarrow \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\ \label{hilb40:26} $26$ & $(C\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:27} $27$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:26]{$26$} } \\ \label{hilb40:28} $28$ & $\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:23]{$23$}, \hyperref[hilb40:27]{$27$}} \\ \label{hilb40:29} $29$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ D)\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:30} $30$ & $(D\ \rightarrow \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ D)\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:29]{$29$} } \\ \label{hilb40:31} $31$ & $(\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb40:30]{$30$} } \\ \label{hilb40:32} $32$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:28]{$28$}, \hyperref[hilb40:31]{$31$}} \\ \label{hilb40:33} $33$ & $(P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\ \label{hilb40:34} $34$ & $(P\ \vee \ B)\ \rightarrow \ (B\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:33]{$33$} } \\ \label{hilb40:35} $35$ & $(C\ \vee \ B)\ \rightarrow \ (B\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:34]{$34$} } \\ \label{hilb40:36} $36$ & $(C\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:35]{$35$} } \\ \label{hilb40:37} $37$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:36]{$36$} } \\ \label{hilb40:38} $38$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \rightarrow \ P)\ \rightarrow \ (A\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\ \label{hilb40:39} $39$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb40:38]{$38$} } \\ \label{hilb40:40} $40$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb40:39]{$39$} } \\ \label{hilb40:41} $41$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ D)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb40:40]{$40$} } \\ \label{hilb40:42} $42$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:43} $43$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:42]{$42$} } \\ \label{hilb40:44} $44$ & $((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:43]{$43$} } \\ \label{hilb40:45} $45$ & $((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:37]{$37$}, \hyperref[hilb40:44]{$44$}} \\ \label{hilb40:46} $46$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:32]{$32$}, \hyperref[hilb40:45]{$45$}} \\ \label{hilb40:47} $47$ & $(C\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:35]{$35$} } \\ \label{hilb40:48} $48$ & $(\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb40:47]{$47$} } \\ \label{hilb40:49} $49$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:50} $50$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:49]{$49$} } \\ \label{hilb40:51} $51$ & $((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb40:50]{$50$} } \\ \label{hilb40:52} $52$ & $((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg \neg (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:46]{$46$}, \hyperref[hilb40:51]{$51$}} \\ \label{hilb40:53} $53$ & $(\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:48]{$48$}, \hyperref[hilb40:52]{$52$}} \\ \label{hilb40:54} $54$ & $(P\ \rightarrow \ B)\ \rightarrow \ (\neg P\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:24]{$24$} } \\ \label{hilb40:55} $55$ & $(C\ \rightarrow \ B)\ \rightarrow \ (\neg C\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:54]{$54$} } \\ \label{hilb40:56} $56$ & $(C\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg C\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:55]{$55$} } \\ \label{hilb40:57} $57$ & $(\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb40:56]{$56$} } \\ \label{hilb40:58} $58$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:59} $59$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:58]{$58$} } \\ \label{hilb40:60} $60$ & $((\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:59]{$59$} } \\ \label{hilb40:61} $61$ & $((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:53]{$53$}, \hyperref[hilb40:60]{$60$}} \\ \label{hilb40:62} $62$ & $(\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:57]{$57$}, \hyperref[hilb40:61]{$61$}} \\ \label{hilb40:63} $63$ & $(\neg P\ \vee \ B)\ \rightarrow \ (P\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:25]{$25$} } \\ \label{hilb40:64} $64$ & $(\neg C\ \vee \ B)\ \rightarrow \ (C\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:63]{$63$} } \\ \label{hilb40:65} $65$ & $(\neg C\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (C\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:64]{$64$} } \\ \label{hilb40:66} $66$ & $(\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:65]{$65$} } \\ \label{hilb40:67} $67$ & $(D\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:58]{$58$} } \\ \label{hilb40:68} $68$ & $((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:67]{$67$} } \\ \label{hilb40:69} $69$ & $((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:66]{$66$}, \hyperref[hilb40:68]{$68$}} \\ \label{hilb40:70} $70$ & $(\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:62]{$62$}, \hyperref[hilb40:69]{$69$}} \\ \label{hilb40:71} $71$ & $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:7]{$7$}, \hyperref[hilb40:70]{$70$}} \\ \label{hilb40:72} $72$ & $\neg \neg P\ \rightarrow \ P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb6}{hilb6} } \\ \label{hilb40:73} $73$ & $\neg \neg A\ \rightarrow \ A$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $A$ in \hyperref[hilb40:72]{$72$} } \\ \label{hilb40:74} $74$ & $\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb40:73]{$73$} } \\ \label{hilb40:75} $75$ & $(P\ \wedge \ Q)\ \rightarrow \ \neg (\neg P\ \vee \ \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand1}{defand1} } \\ \label{hilb40:76} $76$ & $\neg (\neg P\ \vee \ \neg Q)\ \rightarrow \ (P\ \wedge \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand2}{defand2} } \\ \label{hilb40:77} $77$ & $(B\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb40:4]{$4$} } \\ \label{hilb40:78} $78$ & $(\neg \neg (P\ \vee \ Q)\ \rightarrow \ (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:77]{$77$} } \\ \label{hilb40:79} $79$ & $\neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:74]{$74$}, \hyperref[hilb40:78]{$78$}} \\ \label{hilb40:80} $80$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:81} $81$ & $(D\ \rightarrow \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:80]{$80$} } \\ \label{hilb40:82} $82$ & $(\neg (P\ \vee \ Q)\ \rightarrow \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb40:81]{$81$} } \\ \label{hilb40:83} $83$ & $(\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:79]{$79$}, \hyperref[hilb40:82]{$82$}} \\ \label{hilb40:84} $84$ & $(C\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:35]{$35$} } \\ \label{hilb40:85} $85$ & $(\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb40:84]{$84$} } \\ \label{hilb40:86} $86$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:87} $87$ & $(D\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb40:86]{$86$} } \\ \label{hilb40:88} $88$ & $((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:87]{$87$} } \\ \label{hilb40:89} $89$ & $((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:85]{$85$}, \hyperref[hilb40:88]{$88$}} \\ \label{hilb40:90} $90$ & $(\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:83]{$83$}, \hyperref[hilb40:89]{$89$}} \\ \label{hilb40:91} $91$ & $(C\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb40:35]{$35$} } \\ \label{hilb40:92} $92$ & $(\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb40:91]{$91$} } \\ \label{hilb40:93} $93$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:94} $94$ & $(D\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb40:93]{$93$} } \\ \label{hilb40:95} $95$ & $((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)$ in \hyperref[hilb40:94]{$94$} } \\ \label{hilb40:96} $96$ & $((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:90]{$90$}, \hyperref[hilb40:95]{$95$}} \\ \label{hilb40:97} $97$ & $(\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:92]{$92$}, \hyperref[hilb40:96]{$96$}} \\ \label{hilb40:98} $98$ & $(C\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:99} $99$ & $((\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb40:98]{$98$} } \\ \label{hilb40:100} $100$ & $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:97]{$97$}, \hyperref[hilb40:99]{$99$}} \\ \label{hilb40:101} $101$ & $(P\ \wedge \ B)\ \rightarrow \ \neg (\neg P\ \vee \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:75]{$75$} } \\ \label{hilb40:102} $102$ & $(C\ \wedge \ B)\ \rightarrow \ \neg (\neg C\ \vee \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:101]{$101$} } \\ \label{hilb40:103} $103$ & $(C\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg C\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb40:102]{$102$} } \\ \label{hilb40:104} $104$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:103]{$103$} } \\ \label{hilb40:105} $105$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:106} $106$ & $(D\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb40:105]{$105$} } \\ \label{hilb40:107} $107$ & $(\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb40:106]{$106$} } \\ \label{hilb40:108} $108$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:100]{$100$}, \hyperref[hilb40:107]{$107$}} \\ \label{hilb40:109} $109$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:104]{$104$}, \hyperref[hilb40:108]{$108$}} \\ \label{hilb40:110} $110$ & $\neg (\neg P\ \vee \ \neg B)\ \rightarrow \ (P\ \wedge \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:76]{$76$} } \\ \label{hilb40:111} $111$ & $\neg (\neg C\ \vee \ \neg B)\ \rightarrow \ (C\ \wedge \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:110]{$110$} } \\ \label{hilb40:112} $112$ & $\neg (\neg C\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (C\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb40:111]{$111$} } \\ \label{hilb40:113} $113$ & $\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb40:112]{$112$} } \\ \label{hilb40:114} $114$ & $(D\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb40:105]{$105$} } \\ \label{hilb40:115} $115$ & $(\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb40:114]{$114$} } \\ \label{hilb40:116} $116$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:113]{$113$}, \hyperref[hilb40:115]{$115$}} \\ \label{hilb40:117} $117$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:109]{$109$}, \hyperref[hilb40:116]{$116$}} \\ \label{hilb40:118} $118$ & $(C\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:119} $119$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb40:118]{$118$} } \\ \label{hilb40:120} $120$ & $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:117]{$117$}, \hyperref[hilb40:119]{$119$}} \\ \label{hilb40:121} $121$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:122} $122$ & $(D\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:121]{$121$} } \\ \label{hilb40:123} $123$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:122]{$122$} } \\ \label{hilb40:124} $124$ & $(\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:120]{$120$}, \hyperref[hilb40:123]{$123$}} \\ \label{hilb40:125} $125$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:56]{$56$} } \\ \label{hilb40:126} $126$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:127} $127$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:126]{$126$} } \\ \label{hilb40:128} $128$ & $((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:127]{$127$} } \\ \label{hilb40:129} $129$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:124]{$124$}, \hyperref[hilb40:128]{$128$}} \\ \label{hilb40:130} $130$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:125]{$125$}, \hyperref[hilb40:129]{$129$}} \\ \label{hilb40:131} $131$ & $(\neg C\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (C\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:64]{$64$} } \\ \label{hilb40:132} $132$ & $(\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:131]{$131$} } \\ \label{hilb40:133} $133$ & $(D\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:126]{$126$} } \\ \label{hilb40:134} $134$ & $((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:133]{$133$} } \\ \label{hilb40:135} $135$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:132]{$132$}, \hyperref[hilb40:134]{$134$}} \\ \label{hilb40:136} $136$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:130]{$130$}, \hyperref[hilb40:135]{$135$}} \\ \label{hilb40:137} $137$ & $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:71]{$71$}, \hyperref[hilb40:136]{$136$}} \\ \label{hilb40:138} $138$ & $\neg \neg B\ \rightarrow \ B$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb40:72]{$72$} } \\ \label{hilb40:139} $139$ & $\neg \neg (P\ \vee \ A)\ \rightarrow \ (P\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb40:138]{$138$} } \\ \label{hilb40:140} $140$ & $(C\ \rightarrow \ (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:141} $141$ & $(\neg \neg (P\ \vee \ A)\ \rightarrow \ (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \rightarrow \ \neg \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ A)$ in \hyperref[hilb40:140]{$140$} } \\ \label{hilb40:142} $142$ & $\neg (P\ \vee \ A)\ \rightarrow \ \neg \neg \neg (P\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:139]{$139$}, \hyperref[hilb40:141]{$141$}} \\ \label{hilb40:143} $143$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:144} $144$ & $(D\ \rightarrow \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ A)$ in \hyperref[hilb40:143]{$143$} } \\ \label{hilb40:145} $145$ & $(\neg (P\ \vee \ A)\ \rightarrow \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb40:144]{$144$} } \\ \label{hilb40:146} $146$ & $(\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:142]{$142$}, \hyperref[hilb40:145]{$145$}} \\ \label{hilb40:147} $147$ & $(C\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:148} $148$ & $((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb40:147]{$147$} } \\ \label{hilb40:149} $149$ & $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:146]{$146$}, \hyperref[hilb40:148]{$148$}} \\ \label{hilb40:150} $150$ & $(C\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg C\ \vee \ \neg \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ A)$ in \hyperref[hilb40:102]{$102$} } \\ \label{hilb40:151} $151$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:150]{$150$} } \\ \label{hilb40:152} $152$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:153} $153$ & $(D\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb40:152]{$152$} } \\ \label{hilb40:154} $154$ & $(\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))$ in \hyperref[hilb40:153]{$153$} } \\ \label{hilb40:155} $155$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:149]{$149$}, \hyperref[hilb40:154]{$154$}} \\ \label{hilb40:156} $156$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:151]{$151$}, \hyperref[hilb40:155]{$155$}} \\ \label{hilb40:157} $157$ & $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb40:112]{$112$} } \\ \label{hilb40:158} $158$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb40:152]{$152$} } \\ \label{hilb40:159} $159$ & $(\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb40:158]{$158$} } \\ \label{hilb40:160} $160$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:157]{$157$}, \hyperref[hilb40:159]{$159$}} \\ \label{hilb40:161} $161$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:156]{$156$}, \hyperref[hilb40:160]{$160$}} \\ \label{hilb40:162} $162$ & $(C\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:163} $163$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)$ in \hyperref[hilb40:162]{$162$} } \\ \label{hilb40:164} $164$ & $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:161]{$161$}, \hyperref[hilb40:163]{$163$}} \\ \label{hilb40:165} $165$ & $(D\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb40:121]{$121$} } \\ \label{hilb40:166} $166$ & $(\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:165]{$165$} } \\ \label{hilb40:167} $167$ & $(\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:164]{$164$}, \hyperref[hilb40:166]{$166$}} \\ \label{hilb40:168} $168$ & $(C\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg C\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:55]{$55$} } \\ \label{hilb40:169} $169$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:168]{$168$} } \\ \label{hilb40:170} $170$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:171} $171$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb40:170]{$170$} } \\ \label{hilb40:172} $172$ & $((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb40:171]{$171$} } \\ \label{hilb40:173} $173$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:167]{$167$}, \hyperref[hilb40:172]{$172$}} \\ \label{hilb40:174} $174$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:169]{$169$}, \hyperref[hilb40:173]{$173$}} \\ \label{hilb40:175} $175$ & $(\neg C\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (C\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb40:64]{$64$} } \\ \label{hilb40:176} $176$ & $(\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb40:175]{$175$} } \\ \label{hilb40:177} $177$ & $(D\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb40:170]{$170$} } \\ \label{hilb40:178} $178$ & $((\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb40:177]{$177$} } \\ \label{hilb40:179} $179$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:176]{$176$}, \hyperref[hilb40:178]{$178$}} \\ \label{hilb40:180} $180$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:174]{$174$}, \hyperref[hilb40:179]{$179$}} \\ \label{hilb40:181} $181$ & $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:137]{$137$}, \hyperref[hilb40:180]{$180$}} \\ \label{hilb40:182} $182$ & $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ B))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb40:181]{$181$} } \\ \label{hilb40:183} $183$ & $\neg (P\ \vee \ \neg \neg (C\ \wedge \ B))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ C)\ \wedge \ \neg \neg (P\ \vee \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb40:182]{$182$} } \\ \label{hilb40:184} $184$ & $\neg (D\ \vee \ \neg \neg (C\ \wedge \ B))\ \rightarrow \ \neg (\neg \neg (D\ \vee \ C)\ \wedge \ \neg \neg (D\ \vee \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb40:183]{$183$} } \\ \label{hilb40:185} $185$ & $\neg (D\ \vee \ \neg \neg (C\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (D\ \vee \ C)\ \wedge \ \neg \neg (D\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg A$ in \hyperref[hilb40:184]{$184$} } \\ \label{hilb40:186} $186$ & $\neg (D\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (D\ \vee \ \neg Q)\ \wedge \ \neg \neg (D\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg Q$ in \hyperref[hilb40:185]{$185$} } \\ \label{hilb40:187} $187$ & $\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg P$ in \hyperref[hilb40:186]{$186$} } \\ \label{hilb40:188} $188$ & $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb40:187]{$187$} at occurence $1$ } \\ \label{hilb40:189} $189$ & $(P\ \vee \ Q)\ \rightarrow \ \neg (\neg P\ \wedge \ \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb38}{hilb38} } \\ \label{hilb40:190} $190$ & $(P\ \vee \ B)\ \rightarrow \ \neg (\neg P\ \wedge \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:189]{$189$} } \\ \label{hilb40:191} $191$ & $(C\ \vee \ B)\ \rightarrow \ \neg (\neg C\ \wedge \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:190]{$190$} } \\ \label{hilb40:192} $192$ & $(C\ \vee \ A)\ \rightarrow \ \neg (\neg C\ \wedge \ \neg A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $A$ in \hyperref[hilb40:191]{$191$} } \\ \label{hilb40:193} $193$ & $(Q\ \vee \ A)\ \rightarrow \ \neg (\neg Q\ \wedge \ \neg A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q$ in \hyperref[hilb40:192]{$192$} } \\ \label{hilb40:194} $194$ & $(C\ \rightarrow \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (\neg \neg (\neg Q\ \wedge \ \neg A)\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:195} $195$ & $((Q\ \vee \ A)\ \rightarrow \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (\neg \neg (\neg Q\ \wedge \ \neg A)\ \rightarrow \ \neg (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q\ \vee \ A$ in \hyperref[hilb40:194]{$194$} } \\ \label{hilb40:196} $196$ & $\neg \neg (\neg Q\ \wedge \ \neg A)\ \rightarrow \ \neg (Q\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:193]{$193$}, \hyperref[hilb40:195]{$195$}} \\ \label{hilb40:197} $197$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg P\ \vee \ D)\ \rightarrow \ (\neg P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg P$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:198} $198$ & $(D\ \rightarrow \ \neg (Q\ \vee \ A))\ \rightarrow \ ((\neg P\ \vee \ D)\ \rightarrow \ (\neg P\ \vee \ \neg (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (Q\ \vee \ A)$ in \hyperref[hilb40:197]{$197$} } \\ \label{hilb40:199} $199$ & $(\neg \neg (\neg Q\ \wedge \ \neg A)\ \rightarrow \ \neg (Q\ \vee \ A))\ \rightarrow \ ((\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (\neg P\ \vee \ \neg (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb40:198]{$198$} } \\ \label{hilb40:200} $200$ & $(\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (\neg P\ \vee \ \neg (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:196]{$196$}, \hyperref[hilb40:199]{$199$}} \\ \label{hilb40:201} $201$ & $(C\ \rightarrow \ (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg P\ \vee \ \neg (Q\ \vee \ A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:202} $202$ & $((\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb40:201]{$201$} } \\ \label{hilb40:203} $203$ & $\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:200]{$200$}, \hyperref[hilb40:202]{$202$}} \\ \label{hilb40:204} $204$ & $(C\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg C\ \vee \ \neg (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \vee \ A$ in \hyperref[hilb40:102]{$102$} } \\ \label{hilb40:205} $205$ & $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P$ in \hyperref[hilb40:204]{$204$} } \\ \label{hilb40:206} $206$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:207} $207$ & $(D\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb40:206]{$206$} } \\ \label{hilb40:208} $208$ & $(\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))$ in \hyperref[hilb40:207]{$207$} } \\ \label{hilb40:209} $209$ & $((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:203]{$203$}, \hyperref[hilb40:208]{$208$}} \\ \label{hilb40:210} $210$ & $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:205]{$205$}, \hyperref[hilb40:209]{$209$}} \\ \label{hilb40:211} $211$ & $\neg (\neg C\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (C\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb40:111]{$111$} } \\ \label{hilb40:212} $212$ & $\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P$ in \hyperref[hilb40:211]{$211$} } \\ \label{hilb40:213} $213$ & $(D\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb40:206]{$206$} } \\ \label{hilb40:214} $214$ & $(\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb40:213]{$213$} } \\ \label{hilb40:215} $215$ & $((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:212]{$212$}, \hyperref[hilb40:214]{$214$}} \\ \label{hilb40:216} $216$ & $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:210]{$210$}, \hyperref[hilb40:215]{$215$}} \\ \label{hilb40:217} $217$ & $(C\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb40:20]{$20$} } \\ \label{hilb40:218} $218$ & $((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb40:217]{$217$} } \\ \label{hilb40:219} $219$ & $\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (P\ \wedge \ (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:216]{$216$}, \hyperref[hilb40:218]{$218$}} \\ \label{hilb40:220} $220$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ D)\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:221} $221$ & $(D\ \rightarrow \ \neg (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ ((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ D)\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \wedge \ (Q\ \vee \ A))$ in \hyperref[hilb40:220]{$220$} } \\ \label{hilb40:222} $222$ & $(\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ ((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb40:221]{$221$} } \\ \label{hilb40:223} $223$ & $(\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:219]{$219$}, \hyperref[hilb40:222]{$222$}} \\ \label{hilb40:224} $224$ & $(C\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \wedge \ (Q\ \vee \ A))$ in \hyperref[hilb40:35]{$35$} } \\ \label{hilb40:225} $225$ & $(\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:224]{$224$} } \\ \label{hilb40:226} $226$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:227} $227$ & $(D\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:226]{$226$} } \\ \label{hilb40:228} $228$ & $((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ ((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A))$ in \hyperref[hilb40:227]{$227$} } \\ \label{hilb40:229} $229$ & $((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ ((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:225]{$225$}, \hyperref[hilb40:228]{$228$}} \\ \label{hilb40:230} $230$ & $(\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:223]{$223$}, \hyperref[hilb40:229]{$229$}} \\ \label{hilb40:231} $231$ & $(C\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:35]{$35$} } \\ \label{hilb40:232} $232$ & $(\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb40:231]{$231$} } \\ \label{hilb40:233} $233$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:234} $234$ & $(D\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:233]{$233$} } \\ \label{hilb40:235} $235$ & $((\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb40:234]{$234$} } \\ \label{hilb40:236} $236$ & $((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \vee \ \neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:230]{$230$}, \hyperref[hilb40:235]{$235$}} \\ \label{hilb40:237} $237$ & $(\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:232]{$232$}, \hyperref[hilb40:236]{$236$}} \\ \label{hilb40:238} $238$ & $(C\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg C\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:55]{$55$} } \\ \label{hilb40:239} $239$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb40:238]{$238$} } \\ \label{hilb40:240} $240$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:241} $241$ & $(D\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:240]{$240$} } \\ \label{hilb40:242} $242$ & $((\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:241]{$241$} } \\ \label{hilb40:243} $243$ & $(((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:237]{$237$}, \hyperref[hilb40:242]{$242$}} \\ \label{hilb40:244} $244$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:239]{$239$}, \hyperref[hilb40:243]{$243$}} \\ \label{hilb40:245} $245$ & $(\neg C\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (C\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:64]{$64$} } \\ \label{hilb40:246} $246$ & $(\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb40:245]{$245$} } \\ \label{hilb40:247} $247$ & $(D\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:240]{$240$} } \\ \label{hilb40:248} $248$ & $((\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:247]{$247$} } \\ \label{hilb40:249} $249$ & $(((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:246]{$246$}, \hyperref[hilb40:248]{$248$}} \\ \label{hilb40:250} $250$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:244]{$244$}, \hyperref[hilb40:249]{$249$}} \\ \label{hilb40:251} $251$ & $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:188]{$188$}, \hyperref[hilb40:250]{$250$}} \\ \label{hilb40:252} $252$ & $\neg (\neg P\ \wedge \ \neg Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb39}{hilb39} } \\ \label{hilb40:253} $253$ & $\neg (\neg P\ \wedge \ \neg B)\ \rightarrow \ (P\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb40:252]{$252$} } \\ \label{hilb40:254} $254$ & $\neg (\neg C\ \wedge \ \neg B)\ \rightarrow \ (C\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb40:253]{$253$} } \\ \label{hilb40:255} $255$ & $\neg (\neg C\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \rightarrow \ (C\ \vee \ \neg (\neg P\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg P\ \vee \ \neg A)$ in \hyperref[hilb40:254]{$254$} } \\ \label{hilb40:256} $256$ & $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg P\ \vee \ \neg Q)$ in \hyperref[hilb40:255]{$255$} } \\ \label{hilb40:257} $257$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ D)\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \wedge \ (Q\ \vee \ A))$ in \hyperref[hilb40:14]{$14$} } \\ \label{hilb40:258} $258$ & $(D\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ D)\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)$ in \hyperref[hilb40:257]{$257$} } \\ \label{hilb40:259} $259$ & $(\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:258]{$258$} } \\ \label{hilb40:260} $260$ & $(\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:256]{$256$}, \hyperref[hilb40:259]{$259$}} \\ \label{hilb40:261} $261$ & $((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb40:238]{$238$} } \\ \label{hilb40:262} $262$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:41]{$41$} } \\ \label{hilb40:263} $263$ & $(D\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:262]{$262$} } \\ \label{hilb40:264} $264$ & $((\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:263]{$263$} } \\ \label{hilb40:265} $265$ & $(((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:260]{$260$}, \hyperref[hilb40:264]{$264$}} \\ \label{hilb40:266} $266$ & $((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:261]{$261$}, \hyperref[hilb40:265]{$265$}} \\ \label{hilb40:267} $267$ & $(\neg C\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (C\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)$ in \hyperref[hilb40:64]{$64$} } \\ \label{hilb40:268} $268$ & $(\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb40:267]{$267$} } \\ \label{hilb40:269} $269$ & $(D\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:262]{$262$} } \\ \label{hilb40:270} $270$ & $((\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))$ in \hyperref[hilb40:269]{$269$} } \\ \label{hilb40:271} $271$ & $(((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ (\neg (P\ \wedge \ (Q\ \vee \ A))\ \vee \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))\ \rightarrow \ (((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:268]{$268$}, \hyperref[hilb40:270]{$270$}} \\ \label{hilb40:272} $272$ & $((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A)))\ \rightarrow \ ((P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:266]{$266$}, \hyperref[hilb40:271]{$271$}} \\ \label{hilb40:273} $273$ & $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:251]{$251$}, \hyperref[hilb40:272]{$272$}} \\ \label{hilb40:274} $274$ & $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ ((P\ \wedge \ Q)\ \vee \ \neg (\neg P\ \vee \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb40:273]{$273$} at occurence $1$ } \\ \label{hilb40:275} $275$ & $(P\ \wedge \ (Q\ \vee \ A))\ \rightarrow \ ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb40:274]{$274$} at occurence $1$ } \\ & & \qedhere \end{longtable} \end{proof} The second distributive law (second direction): \begin{thm}[hilb41] \hypertarget{hilb41}{} \begin{displaymath} ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))\end{displaymath} \end{thm} \begin{proof} \mbox{}\\ \begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}} \label{hilb41:1} $1$ & $(P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb36}{hilb36} } \\ \label{hilb41:2} $2$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg Q\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\ \label{hilb41:3} $3$ & $(P\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $A$ in \hyperref[hilb41:2]{$2$} } \\ \label{hilb41:4} $4$ & $(B\ \rightarrow \ A)\ \rightarrow \ (\neg A\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb41:3]{$3$} } \\ \label{hilb41:5} $5$ & $(B\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb41:4]{$4$} } \\ \label{hilb41:6} $6$ & $((P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb41:5]{$5$} } \\ \label{hilb41:7} $7$ & $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:1]{$1$}, \hyperref[hilb41:6]{$6$}} \\ \label{hilb41:8} $8$ & $\neg \neg P\ \rightarrow \ P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb6}{hilb6} } \\ \label{hilb41:9} $9$ & $\neg \neg B\ \rightarrow \ B$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb41:8]{$8$} } \\ \label{hilb41:10} $10$ & $\neg \neg (Q\ \wedge \ A)\ \rightarrow \ (Q\ \wedge \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \wedge \ A$ in \hyperref[hilb41:9]{$9$} } \\ \label{hilb41:11} $11$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \vee \ P)\ \rightarrow \ (A\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\ \label{hilb41:12} $12$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb41:11]{$11$} } \\ \label{hilb41:13} $13$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ P)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb41:12]{$12$} } \\ \label{hilb41:14} $14$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \vee \ D)\ \rightarrow \ (B\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb41:13]{$13$} } \\ \label{hilb41:15} $15$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:16} $16$ & $(D\ \rightarrow \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ D)\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q\ \wedge \ A$ in \hyperref[hilb41:15]{$15$} } \\ \label{hilb41:17} $17$ & $(\neg \neg (Q\ \wedge \ A)\ \rightarrow \ (Q\ \wedge \ A))\ \rightarrow \ ((P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (Q\ \wedge \ A)$ in \hyperref[hilb41:16]{$16$} } \\ \label{hilb41:18} $18$ & $(P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:10]{$10$}, \hyperref[hilb41:17]{$17$}} \\ \label{hilb41:19} $19$ & $(P\ \rightarrow \ B)\ \rightarrow \ (\neg B\ \rightarrow \ \neg P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:2]{$2$} } \\ \label{hilb41:20} $20$ & $(C\ \rightarrow \ B)\ \rightarrow \ (\neg B\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:19]{$19$} } \\ \label{hilb41:21} $21$ & $(C\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ (Q\ \wedge \ A)$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:22} $22$ & $((P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \rightarrow \ (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ \neg \neg (Q\ \wedge \ A)$ in \hyperref[hilb41:21]{$21$} } \\ \label{hilb41:23} $23$ & $\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:18]{$18$}, \hyperref[hilb41:22]{$22$}} \\ \label{hilb41:24} $24$ & $(P\ \rightarrow \ Q)\ \rightarrow \ (\neg P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\ \label{hilb41:25} $25$ & $(\neg P\ \vee \ Q)\ \rightarrow \ (P\ \rightarrow \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\ \label{hilb41:26} $26$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ D)\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:27} $27$ & $(D\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ ((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ D)\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:26]{$26$} } \\ \label{hilb41:28} $28$ & $(\neg (P\ \vee \ (Q\ \wedge \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ ((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb41:27]{$27$} } \\ \label{hilb41:29} $29$ & $(\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:23]{$23$}, \hyperref[hilb41:28]{$28$}} \\ \label{hilb41:30} $30$ & $(P\ \rightarrow \ B)\ \rightarrow \ (\neg P\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:24]{$24$} } \\ \label{hilb41:31} $31$ & $(C\ \rightarrow \ B)\ \rightarrow \ (\neg C\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:30]{$30$} } \\ \label{hilb41:32} $32$ & $(C\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg C\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb41:31]{$31$} } \\ \label{hilb41:33} $33$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:32]{$32$} } \\ \label{hilb41:34} $34$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((A\ \rightarrow \ P)\ \rightarrow \ (A\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\ \label{hilb41:35} $35$ & $(P\ \rightarrow \ Q)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb41:34]{$34$} } \\ \label{hilb41:36} $36$ & $(P\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ P)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb41:35]{$35$} } \\ \label{hilb41:37} $37$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((B\ \rightarrow \ D)\ \rightarrow \ (B\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb41:36]{$36$} } \\ \label{hilb41:38} $38$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:39} $39$ & $(D\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:38]{$38$} } \\ \label{hilb41:40} $40$ & $((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A))$ in \hyperref[hilb41:39]{$39$} } \\ \label{hilb41:41} $41$ & $((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:29]{$29$}, \hyperref[hilb41:40]{$40$}} \\ \label{hilb41:42} $42$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:33]{$33$}, \hyperref[hilb41:41]{$41$}} \\ \label{hilb41:43} $43$ & $(\neg P\ \vee \ B)\ \rightarrow \ (P\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:25]{$25$} } \\ \label{hilb41:44} $44$ & $(\neg C\ \vee \ B)\ \rightarrow \ (C\ \rightarrow \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:43]{$43$} } \\ \label{hilb41:45} $45$ & $(\neg C\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (C\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:44]{$44$} } \\ \label{hilb41:46} $46$ & $(\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:45]{$45$} } \\ \label{hilb41:47} $47$ & $(D\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:38]{$38$} } \\ \label{hilb41:48} $48$ & $((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:47]{$47$} } \\ \label{hilb41:49} $49$ & $((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:46]{$46$}, \hyperref[hilb41:48]{$48$}} \\ \label{hilb41:50} $50$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ (Q\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:42]{$42$}, \hyperref[hilb41:49]{$49$}} \\ \label{hilb41:51} $51$ & $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:7]{$7$}, \hyperref[hilb41:50]{$50$}} \\ \label{hilb41:52} $52$ & $P\ \rightarrow \ \neg \neg P$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb5}{hilb5} } \\ \label{hilb41:53} $53$ & $A\ \rightarrow \ \neg \neg A$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $A$ in \hyperref[hilb41:52]{$52$} } \\ \label{hilb41:54} $54$ & $(P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $P\ \vee \ Q$ in \hyperref[hilb41:53]{$53$} } \\ \label{hilb41:55} $55$ & $(P\ \wedge \ Q)\ \rightarrow \ \neg (\neg P\ \vee \ \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand1}{defand1} } \\ \label{hilb41:56} $56$ & $\neg (\neg P\ \vee \ \neg Q)\ \rightarrow \ (P\ \wedge \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand2}{defand2} } \\ \label{hilb41:57} $57$ & $(B\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:4]{$4$} } \\ \label{hilb41:58} $58$ & $((P\ \vee \ Q)\ \rightarrow \ \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ Q$ in \hyperref[hilb41:57]{$57$} } \\ \label{hilb41:59} $59$ & $\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:54]{$54$}, \hyperref[hilb41:58]{$58$}} \\ \label{hilb41:60} $60$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:61} $61$ & $(D\ \rightarrow \ \neg (P\ \vee \ Q))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb41:60]{$60$} } \\ \label{hilb41:62} $62$ & $(\neg \neg \neg (P\ \vee \ Q)\ \rightarrow \ \neg (P\ \vee \ Q))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:61]{$61$} } \\ \label{hilb41:63} $63$ & $(\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:59]{$59$}, \hyperref[hilb41:62]{$62$}} \\ \label{hilb41:64} $64$ & $(P\ \vee \ Q)\ \rightarrow \ (Q\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\ \label{hilb41:65} $65$ & $(P\ \vee \ B)\ \rightarrow \ (B\ \vee \ P)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:64]{$64$} } \\ \label{hilb41:66} $66$ & $(C\ \vee \ B)\ \rightarrow \ (B\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:65]{$65$} } \\ \label{hilb41:67} $67$ & $(C\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ Q)$ in \hyperref[hilb41:66]{$66$} } \\ \label{hilb41:68} $68$ & $(\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb41:67]{$67$} } \\ \label{hilb41:69} $69$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:70} $70$ & $(D\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb41:69]{$69$} } \\ \label{hilb41:71} $71$ & $((\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)$ in \hyperref[hilb41:70]{$70$} } \\ \label{hilb41:72} $72$ & $((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:68]{$68$}, \hyperref[hilb41:71]{$71$}} \\ \label{hilb41:73} $73$ & $(\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:63]{$63$}, \hyperref[hilb41:72]{$72$}} \\ \label{hilb41:74} $74$ & $(C\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb41:66]{$66$} } \\ \label{hilb41:75} $75$ & $(\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:74]{$74$} } \\ \label{hilb41:76} $76$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:77} $77$ & $(D\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb41:76]{$76$} } \\ \label{hilb41:78} $78$ & $((\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:77]{$77$} } \\ \label{hilb41:79} $79$ & $((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ A)\ \vee \ \neg \neg \neg (P\ \vee \ Q)))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:73]{$73$}, \hyperref[hilb41:78]{$78$}} \\ \label{hilb41:80} $80$ & $(\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:75]{$75$}, \hyperref[hilb41:79]{$79$}} \\ \label{hilb41:81} $81$ & $(C\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:82} $82$ & $((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb41:81]{$81$} } \\ \label{hilb41:83} $83$ & $\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:80]{$80$}, \hyperref[hilb41:82]{$82$}} \\ \label{hilb41:84} $84$ & $(P\ \wedge \ B)\ \rightarrow \ \neg (\neg P\ \vee \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:55]{$55$} } \\ \label{hilb41:85} $85$ & $(C\ \wedge \ B)\ \rightarrow \ \neg (\neg C\ \vee \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:84]{$84$} } \\ \label{hilb41:86} $86$ & $(C\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg C\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb41:85]{$85$} } \\ \label{hilb41:87} $87$ & $((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ Q$ in \hyperref[hilb41:86]{$86$} } \\ \label{hilb41:88} $88$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:89} $89$ & $(D\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb41:88]{$88$} } \\ \label{hilb41:90} $90$ & $(\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb41:89]{$89$} } \\ \label{hilb41:91} $91$ & $(((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:83]{$83$}, \hyperref[hilb41:90]{$90$}} \\ \label{hilb41:92} $92$ & $((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:87]{$87$}, \hyperref[hilb41:91]{$91$}} \\ \label{hilb41:93} $93$ & $\neg (\neg P\ \vee \ \neg B)\ \rightarrow \ (P\ \wedge \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:56]{$56$} } \\ \label{hilb41:94} $94$ & $\neg (\neg C\ \vee \ \neg B)\ \rightarrow \ (C\ \wedge \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:93]{$93$} } \\ \label{hilb41:95} $95$ & $\neg (\neg C\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (C\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb41:94]{$94$} } \\ \label{hilb41:96} $96$ & $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:95]{$95$} } \\ \label{hilb41:97} $97$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ ((((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb41:88]{$88$} } \\ \label{hilb41:98} $98$ & $(\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ ((((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb41:97]{$97$} } \\ \label{hilb41:99} $99$ & $(((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:96]{$96$}, \hyperref[hilb41:98]{$98$}} \\ \label{hilb41:100} $100$ & $((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:92]{$92$}, \hyperref[hilb41:99]{$99$}} \\ \label{hilb41:101} $101$ & $(C\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:102} $102$ & $(((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb41:101]{$101$} } \\ \label{hilb41:103} $103$ & $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:100]{$100$}, \hyperref[hilb41:102]{$102$}} \\ \label{hilb41:104} $104$ & $(C\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:105} $105$ & $(\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:104]{$104$} } \\ \label{hilb41:106} $106$ & $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:103]{$103$}, \hyperref[hilb41:105]{$105$}} \\ \label{hilb41:107} $107$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:108} $108$ & $(D\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:107]{$107$} } \\ \label{hilb41:109} $109$ & $(\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:108]{$108$} } \\ \label{hilb41:110} $110$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:106]{$106$}, \hyperref[hilb41:109]{$109$}} \\ \label{hilb41:111} $111$ & $(C\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:66]{$66$} } \\ \label{hilb41:112} $112$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:111]{$111$} } \\ \label{hilb41:113} $113$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:114} $114$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:113]{$113$} } \\ \label{hilb41:115} $115$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:114]{$114$} } \\ \label{hilb41:116} $116$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:112]{$112$}, \hyperref[hilb41:115]{$115$}} \\ \label{hilb41:117} $117$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:110]{$110$}, \hyperref[hilb41:116]{$116$}} \\ \label{hilb41:118} $118$ & $(C\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:66]{$66$} } \\ \label{hilb41:119} $119$ & $(\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:118]{$118$} } \\ \label{hilb41:120} $120$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:121} $121$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:120]{$120$} } \\ \label{hilb41:122} $122$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:121]{$121$} } \\ \label{hilb41:123} $123$ & $((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:117]{$117$}, \hyperref[hilb41:122]{$122$}} \\ \label{hilb41:124} $124$ & $(\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:119]{$119$}, \hyperref[hilb41:123]{$123$}} \\ \label{hilb41:125} $125$ & $(C\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg C\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:31]{$31$} } \\ \label{hilb41:126} $126$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:125]{$125$} } \\ \label{hilb41:127} $127$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:128} $128$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:127]{$127$} } \\ \label{hilb41:129} $129$ & $((\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:128]{$128$} } \\ \label{hilb41:130} $130$ & $((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:124]{$124$}, \hyperref[hilb41:129]{$129$}} \\ \label{hilb41:131} $131$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:126]{$126$}, \hyperref[hilb41:130]{$130$}} \\ \label{hilb41:132} $132$ & $(\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:45]{$45$} } \\ \label{hilb41:133} $133$ & $(D\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:127]{$127$} } \\ \label{hilb41:134} $134$ & $((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:133]{$133$} } \\ \label{hilb41:135} $135$ & $((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:132]{$132$}, \hyperref[hilb41:134]{$134$}} \\ \label{hilb41:136} $136$ & $(\neg ((P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:131]{$131$}, \hyperref[hilb41:135]{$135$}} \\ \label{hilb41:137} $137$ & $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:51]{$51$}, \hyperref[hilb41:136]{$136$}} \\ \label{hilb41:138} $138$ & $B\ \rightarrow \ \neg \neg B$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $B$ in \hyperref[hilb41:52]{$52$} } \\ \label{hilb41:139} $139$ & $(P\ \vee \ A)\ \rightarrow \ \neg \neg (P\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \vee \ A$ in \hyperref[hilb41:138]{$138$} } \\ \label{hilb41:140} $140$ & $(C\ \rightarrow \ \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ A)\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ A)$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:141} $141$ & $((P\ \vee \ A)\ \rightarrow \ \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ A)\ \rightarrow \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \vee \ A$ in \hyperref[hilb41:140]{$140$} } \\ \label{hilb41:142} $142$ & $\neg \neg \neg (P\ \vee \ A)\ \rightarrow \ \neg (P\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:139]{$139$}, \hyperref[hilb41:141]{$141$}} \\ \label{hilb41:143} $143$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:144} $144$ & $(D\ \rightarrow \ \neg (P\ \vee \ A))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ D)\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ A)$ in \hyperref[hilb41:143]{$143$} } \\ \label{hilb41:145} $145$ & $(\neg \neg \neg (P\ \vee \ A)\ \rightarrow \ \neg (P\ \vee \ A))\ \rightarrow \ ((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg \neg (P\ \vee \ A)$ in \hyperref[hilb41:144]{$144$} } \\ \label{hilb41:146} $146$ & $(\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:142]{$142$}, \hyperref[hilb41:145]{$145$}} \\ \label{hilb41:147} $147$ & $(C\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:148} $148$ & $((\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)$ in \hyperref[hilb41:147]{$147$} } \\ \label{hilb41:149} $149$ & $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:146]{$146$}, \hyperref[hilb41:148]{$148$}} \\ \label{hilb41:150} $150$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:86]{$86$} } \\ \label{hilb41:151} $151$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:152} $152$ & $(D\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))$ in \hyperref[hilb41:151]{$151$} } \\ \label{hilb41:153} $153$ & $(\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A))$ in \hyperref[hilb41:152]{$152$} } \\ \label{hilb41:154} $154$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:149]{$149$}, \hyperref[hilb41:153]{$153$}} \\ \label{hilb41:155} $155$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:150]{$150$}, \hyperref[hilb41:154]{$154$}} \\ \label{hilb41:156} $156$ & $\neg (\neg C\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ (C\ \wedge \ \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ A)$ in \hyperref[hilb41:94]{$94$} } \\ \label{hilb41:157} $157$ & $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)$ in \hyperref[hilb41:156]{$156$} } \\ \label{hilb41:158} $158$ & $(D\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)$ in \hyperref[hilb41:151]{$151$} } \\ \label{hilb41:159} $159$ & $(\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A))$ in \hyperref[hilb41:158]{$158$} } \\ \label{hilb41:160} $160$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg \neg (P\ \vee \ Q)\ \vee \ \neg \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:157]{$157$}, \hyperref[hilb41:159]{$159$}} \\ \label{hilb41:161} $161$ & $(\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:155]{$155$}, \hyperref[hilb41:160]{$160$}} \\ \label{hilb41:162} $162$ & $(C\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:163} $163$ & $((\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)$ in \hyperref[hilb41:162]{$162$} } \\ \label{hilb41:164} $164$ & $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:161]{$161$}, \hyperref[hilb41:163]{$163$}} \\ \label{hilb41:165} $165$ & $(C\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:166} $166$ & $(\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb41:165]{$165$} } \\ \label{hilb41:167} $167$ & $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:164]{$164$}, \hyperref[hilb41:166]{$166$}} \\ \label{hilb41:168} $168$ & $(D\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb41:107]{$107$} } \\ \label{hilb41:169} $169$ & $(\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:168]{$168$} } \\ \label{hilb41:170} $170$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:167]{$167$}, \hyperref[hilb41:169]{$169$}} \\ \label{hilb41:171} $171$ & $(C\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb41:66]{$66$} } \\ \label{hilb41:172} $172$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:171]{$171$} } \\ \label{hilb41:173} $173$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:174} $174$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:173]{$173$} } \\ \label{hilb41:175} $175$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb41:174]{$174$} } \\ \label{hilb41:176} $176$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))))\ \rightarrow \ ((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:172]{$172$}, \hyperref[hilb41:175]{$175$}} \\ \label{hilb41:177} $177$ & $(\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:170]{$170$}, \hyperref[hilb41:176]{$176$}} \\ \label{hilb41:178} $178$ & $(\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:118]{$118$} } \\ \label{hilb41:179} $179$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:180} $180$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:179]{$179$} } \\ \label{hilb41:181} $181$ & $((\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:180]{$180$} } \\ \label{hilb41:182} $182$ & $((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))\ \vee \ \neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))))\ \rightarrow \ ((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:177]{$177$}, \hyperref[hilb41:181]{$181$}} \\ \label{hilb41:183} $183$ & $(\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:178]{$178$}, \hyperref[hilb41:182]{$182$}} \\ \label{hilb41:184} $184$ & $(\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))$ in \hyperref[hilb41:125]{$125$} } \\ \label{hilb41:185} $185$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:186} $186$ & $(D\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:185]{$185$} } \\ \label{hilb41:187} $187$ & $((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:186]{$186$} } \\ \label{hilb41:188} $188$ & $((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:183]{$183$}, \hyperref[hilb41:187]{$187$}} \\ \label{hilb41:189} $189$ & $(\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:184]{$184$}, \hyperref[hilb41:188]{$188$}} \\ \label{hilb41:190} $190$ & $(\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))$ in \hyperref[hilb41:45]{$45$} } \\ \label{hilb41:191} $191$ & $(D\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:185]{$185$} } \\ \label{hilb41:192} $192$ & $((\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ (((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ in \hyperref[hilb41:191]{$191$} } \\ \label{hilb41:193} $193$ & $((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \vee \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))\ \rightarrow \ ((\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:190]{$190$}, \hyperref[hilb41:192]{$192$}} \\ \label{hilb41:194} $194$ & $(\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))\ \rightarrow \ (\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:189]{$189$}, \hyperref[hilb41:193]{$193$}} \\ \label{hilb41:195} $195$ & $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ A))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:137]{$137$}, \hyperref[hilb41:194]{$194$}} \\ \label{hilb41:196} $196$ & $\neg (\neg \neg (P\ \vee \ Q)\ \wedge \ \neg \neg (P\ \vee \ B))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (Q\ \wedge \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $A$ by $B$ in \hyperref[hilb41:195]{$195$} } \\ \label{hilb41:197} $197$ & $\neg (\neg \neg (P\ \vee \ C)\ \wedge \ \neg \neg (P\ \vee \ B))\ \rightarrow \ \neg (P\ \vee \ \neg \neg (C\ \wedge \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $C$ in \hyperref[hilb41:196]{$196$} } \\ \label{hilb41:198} $198$ & $\neg (\neg \neg (D\ \vee \ C)\ \wedge \ \neg \neg (D\ \vee \ B))\ \rightarrow \ \neg (D\ \vee \ \neg \neg (C\ \wedge \ B))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $D$ in \hyperref[hilb41:197]{$197$} } \\ \label{hilb41:199} $199$ & $\neg (\neg \neg (D\ \vee \ C)\ \wedge \ \neg \neg (D\ \vee \ \neg A))\ \rightarrow \ \neg (D\ \vee \ \neg \neg (C\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg A$ in \hyperref[hilb41:198]{$198$} } \\ \label{hilb41:200} $200$ & $\neg (\neg \neg (D\ \vee \ \neg Q)\ \wedge \ \neg \neg (D\ \vee \ \neg A))\ \rightarrow \ \neg (D\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg Q$ in \hyperref[hilb41:199]{$199$} } \\ \label{hilb41:201} $201$ & $\neg (\neg \neg (\neg P\ \vee \ \neg Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg P$ in \hyperref[hilb41:200]{$200$} } \\ \label{hilb41:202} $202$ & $\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg \neg (\neg P\ \vee \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb41:201]{$201$} at occurence $1$ } \\ \label{hilb41:203} $203$ & $\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb41:202]{$202$} at occurence $1$ } \\ \label{hilb41:204} $204$ & $\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb41:203]{$203$} at occurence $1$ } \\ \label{hilb41:205} $205$ & $(P\ \vee \ Q)\ \rightarrow \ \neg (\neg P\ \wedge \ \neg Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb38}{hilb38} } \\ \label{hilb41:206} $206$ & $(P\ \vee \ B)\ \rightarrow \ \neg (\neg P\ \wedge \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:205]{$205$} } \\ \label{hilb41:207} $207$ & $(C\ \vee \ B)\ \rightarrow \ \neg (\neg C\ \wedge \ \neg B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:206]{$206$} } \\ \label{hilb41:208} $208$ & $(C\ \vee \ (P\ \wedge \ A))\ \rightarrow \ \neg (\neg C\ \wedge \ \neg (P\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ A$ in \hyperref[hilb41:207]{$207$} } \\ \label{hilb41:209} $209$ & $((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ Q$ in \hyperref[hilb41:208]{$208$} } \\ \label{hilb41:210} $210$ & $(C\ \rightarrow \ \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:211} $211$ & $(((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ (\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)$ in \hyperref[hilb41:210]{$210$} } \\ \label{hilb41:212} $212$ & $\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:209]{$209$}, \hyperref[hilb41:211]{$211$}} \\ \label{hilb41:213} $213$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ D)\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:214} $214$ & $(D\ \rightarrow \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ D)\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))$ in \hyperref[hilb41:213]{$213$} } \\ \label{hilb41:215} $215$ & $(\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))$ in \hyperref[hilb41:214]{$214$} } \\ \label{hilb41:216} $216$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:212]{$212$}, \hyperref[hilb41:215]{$215$}} \\ \label{hilb41:217} $217$ & $(C\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))$ in \hyperref[hilb41:66]{$66$} } \\ \label{hilb41:218} $218$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:217]{$217$} } \\ \label{hilb41:219} $219$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:220} $220$ & $(D\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ D)\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:219]{$219$} } \\ \label{hilb41:221} $221$ & $(((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))$ in \hyperref[hilb41:220]{$220$} } \\ \label{hilb41:222} $222$ & $(((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:218]{$218$}, \hyperref[hilb41:221]{$221$}} \\ \label{hilb41:223} $223$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:216]{$216$}, \hyperref[hilb41:222]{$222$}} \\ \label{hilb41:224} $224$ & $(C\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:66]{$66$} } \\ \label{hilb41:225} $225$ & $(\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))$ in \hyperref[hilb41:224]{$224$} } \\ \label{hilb41:226} $226$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:227} $227$ & $(D\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ (((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:226]{$226$} } \\ \label{hilb41:228} $228$ & $(((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ (((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))))\ \rightarrow \ ((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))$ in \hyperref[hilb41:227]{$227$} } \\ \label{hilb41:229} $229$ & $((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \vee \ \neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))))\ \rightarrow \ ((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:223]{$223$}, \hyperref[hilb41:228]{$228$}} \\ \label{hilb41:230} $230$ & $(\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:225]{$225$}, \hyperref[hilb41:229]{$229$}} \\ \label{hilb41:231} $231$ & $(C\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg C\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:31]{$31$} } \\ \label{hilb41:232} $232$ & $(\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))$ in \hyperref[hilb41:231]{$231$} } \\ \label{hilb41:233} $233$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:234} $234$ & $(D\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ (((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:233]{$233$} } \\ \label{hilb41:235} $235$ & $((\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ (((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:234]{$234$} } \\ \label{hilb41:236} $236$ & $((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg \neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:230]{$230$}, \hyperref[hilb41:235]{$235$}} \\ \label{hilb41:237} $237$ & $(\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:232]{$232$}, \hyperref[hilb41:236]{$236$}} \\ \label{hilb41:238} $238$ & $(\neg C\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (C\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:44]{$44$} } \\ \label{hilb41:239} $239$ & $(\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)$ in \hyperref[hilb41:238]{$238$} } \\ \label{hilb41:240} $240$ & $(D\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ (((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:233]{$233$} } \\ \label{hilb41:241} $241$ & $((\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ (((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:240]{$240$} } \\ \label{hilb41:242} $242$ & $((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:239]{$239$}, \hyperref[hilb41:241]{$241$}} \\ \label{hilb41:243} $243$ & $(\neg (\neg (P\ \wedge \ Q)\ \wedge \ \neg (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:237]{$237$}, \hyperref[hilb41:242]{$242$}} \\ \label{hilb41:244} $244$ & $((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:204]{$204$}, \hyperref[hilb41:243]{$243$}} \\ \label{hilb41:245} $245$ & $\neg (\neg P\ \wedge \ \neg Q)\ \rightarrow \ (P\ \vee \ Q)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb39}{hilb39} } \\ \label{hilb41:246} $246$ & $\neg (\neg P\ \wedge \ \neg B)\ \rightarrow \ (P\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $Q$ by $B$ in \hyperref[hilb41:245]{$245$} } \\ \label{hilb41:247} $247$ & $\neg (\neg C\ \wedge \ \neg B)\ \rightarrow \ (C\ \vee \ B)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $P$ by $C$ in \hyperref[hilb41:246]{$246$} } \\ \label{hilb41:248} $248$ & $\neg (\neg C\ \wedge \ \neg A)\ \rightarrow \ (C\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $A$ in \hyperref[hilb41:247]{$247$} } \\ \label{hilb41:249} $249$ & $\neg (\neg Q\ \wedge \ \neg A)\ \rightarrow \ (Q\ \vee \ A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $Q$ in \hyperref[hilb41:248]{$248$} } \\ \label{hilb41:250} $250$ & $(C\ \rightarrow \ (Q\ \vee \ A))\ \rightarrow \ (\neg (Q\ \vee \ A)\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \vee \ A$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:251} $251$ & $(\neg (\neg Q\ \wedge \ \neg A)\ \rightarrow \ (Q\ \vee \ A))\ \rightarrow \ (\neg (Q\ \vee \ A)\ \rightarrow \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:250]{$250$} } \\ \label{hilb41:252} $252$ & $\neg (Q\ \vee \ A)\ \rightarrow \ \neg \neg (\neg Q\ \wedge \ \neg A)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:249]{$249$}, \hyperref[hilb41:251]{$251$}} \\ \label{hilb41:253} $253$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg P\ \vee \ D)\ \rightarrow \ (\neg P\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg P$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:254} $254$ & $(D\ \rightarrow \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ ((\neg P\ \vee \ D)\ \rightarrow \ (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:253]{$253$} } \\ \label{hilb41:255} $255$ & $(\neg (Q\ \vee \ A)\ \rightarrow \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ ((\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (Q\ \vee \ A)$ in \hyperref[hilb41:254]{$254$} } \\ \label{hilb41:256} $256$ & $(\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:252]{$252$}, \hyperref[hilb41:255]{$255$}} \\ \label{hilb41:257} $257$ & $(C\ \rightarrow \ (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg C)$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:20]{$20$} } \\ \label{hilb41:258} $258$ & $((\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg P\ \vee \ \neg (Q\ \vee \ A)$ in \hyperref[hilb41:257]{$257$} } \\ \label{hilb41:259} $259$ & $\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:256]{$256$}, \hyperref[hilb41:258]{$258$}} \\ \label{hilb41:260} $260$ & $(C\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg C\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:85]{$85$} } \\ \label{hilb41:261} $261$ & $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P$ in \hyperref[hilb41:260]{$260$} } \\ \label{hilb41:262} $262$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:263} $263$ & $(D\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))$ in \hyperref[hilb41:262]{$262$} } \\ \label{hilb41:264} $264$ & $(\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:263]{$263$} } \\ \label{hilb41:265} $265$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:259]{$259$}, \hyperref[hilb41:264]{$264$}} \\ \label{hilb41:266} $266$ & $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:261]{$261$}, \hyperref[hilb41:265]{$265$}} \\ \label{hilb41:267} $267$ & $\neg (\neg C\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ (C\ \wedge \ (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $Q\ \vee \ A$ in \hyperref[hilb41:94]{$94$} } \\ \label{hilb41:268} $268$ & $\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P$ in \hyperref[hilb41:267]{$267$} } \\ \label{hilb41:269} $269$ & $(D\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ D)\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb41:262]{$262$} } \\ \label{hilb41:270} $270$ & $(\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg (\neg P\ \vee \ \neg (Q\ \vee \ A))$ in \hyperref[hilb41:269]{$269$} } \\ \label{hilb41:271} $271$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ \neg (\neg P\ \vee \ \neg (Q\ \vee \ A)))\ \rightarrow \ ((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:268]{$268$}, \hyperref[hilb41:270]{$270$}} \\ \label{hilb41:272} $272$ & $(P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:266]{$266$}, \hyperref[hilb41:271]{$271$}} \\ \label{hilb41:273} $273$ & $(D\ \rightarrow \ C)\ \rightarrow \ ((\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))$ in \hyperref[hilb41:14]{$14$} } \\ \label{hilb41:274} $274$ & $(D\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ ((\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ D)\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb41:273]{$273$} } \\ \label{hilb41:275} $275$ & $((P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ ((\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)$ in \hyperref[hilb41:274]{$274$} } \\ \label{hilb41:276} $276$ & $(\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:272]{$272$}, \hyperref[hilb41:275]{$275$}} \\ \label{hilb41:277} $277$ & $(((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)$ in \hyperref[hilb41:231]{$231$} } \\ \label{hilb41:278} $278$ & $(D\ \rightarrow \ C)\ \rightarrow \ (((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ C))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:37]{$37$} } \\ \label{hilb41:279} $279$ & $(D\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ (((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))$ in \hyperref[hilb41:278]{$278$} } \\ \label{hilb41:280} $280$ & $((\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ (((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))$ in \hyperref[hilb41:279]{$279$} } \\ \label{hilb41:281} $281$ & $((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A))))\ \rightarrow \ ((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:276]{$276$}, \hyperref[hilb41:280]{$280$}} \\ \label{hilb41:282} $282$ & $(((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:277]{$277$}, \hyperref[hilb41:281]{$281$}} \\ \label{hilb41:283} $283$ & $(\neg C\ \vee \ (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (C\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $B$ by $P\ \wedge \ (Q\ \vee \ A)$ in \hyperref[hilb41:44]{$44$} } \\ \label{hilb41:284} $284$ & $(\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $(P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A)$ in \hyperref[hilb41:283]{$283$} } \\ \label{hilb41:285} $285$ & $(D\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ (((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ D)\ \rightarrow \ ((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $C$ by $((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))$ in \hyperref[hilb41:278]{$278$} } \\ \label{hilb41:286} $286$ & $((\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ (((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ ((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace} $D$ by $\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))$ in \hyperref[hilb41:285]{$285$} } \\ \label{hilb41:287} $287$ & $((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (\neg ((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \vee \ (P\ \wedge \ (Q\ \vee \ A))))\ \rightarrow \ ((((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:284]{$284$}, \hyperref[hilb41:286]{$286$}} \\ \label{hilb41:288} $288$ & $(((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ \neg (\neg Q\ \wedge \ \neg A)))\ \rightarrow \ (((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A)))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:282]{$282$}, \hyperref[hilb41:287]{$287$}} \\ \label{hilb41:289} $289$ & $((P\ \wedge \ Q)\ \vee \ (P\ \wedge \ A))\ \rightarrow \ (P\ \wedge \ (Q\ \vee \ A))$ & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:244]{$244$}, \hyperref[hilb41:288]{$288$}} \\ & & \qedhere \end{longtable} \end{proof} \section{Cross Reference} This module is used by the following modules: \begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}} Name: & predtheo2 \\ Version: & 1.00.00 \\ Rule version: & 1.02.00 \\ Orgin: & \url{predtheo2_1.00.00_1.02.00.qedeq} \\ pdf: & \url{predtheo2_1.00.00_1.02.00.pdf} \\ \end{longtable} \end{document}