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\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}
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\title{Further Theorems of Propositional Calculus}
\author{Michael Meyling}
\date{\tt <module@qedeq.org>}


\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: & prophilbert2 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{http://www.qedeq.org/0_00_53/prophilbert2_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: & prophilbert1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{prophilbert1_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{prophilbert1_1.00.00_1.00.00.pdf}  \\
\end{longtable}


\section*{Content}




Negation of a conjunction:




\begin{thm}[hilb18]
\hypertarget{hilb18}{}
\begin{displaymath}
\neg (P\ \wedge \  Q)\ \rightarrow \  (\neg P\ \vee \  \neg Q)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb18:1}
  $1$ & $\neg \neg P\ \rightarrow \  P$
  & {\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{hilb18:2}
  $2$ & $\neg \neg Q\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb18:1]{$1$}
} \\
\label{hilb18:3}
  $3$ & $\neg \neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb18:2]{$2$}
} \\
\label{hilb18:4}
  $4$ & $\neg (P\ \wedge \  Q)\ \rightarrow \  (\neg P\ \vee \  \neg Q)$
  & {\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[hilb18:3]{$3$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



The reverse of a negation of a conjunction:




\begin{thm}[hilb19]
\hypertarget{hilb19}{}
\begin{displaymath}
(\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg (P\ \wedge \  Q)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb19:1}
  $1$ & $P\ \rightarrow \  \neg \neg P$
  & {\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{hilb19:2}
  $2$ & $Q\ \rightarrow \  \neg \neg Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb19:1]{$1$}
} \\
\label{hilb19:3}
  $3$ & $(\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg \neg (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb19:2]{$2$}
} \\
\label{hilb19:4}
  $4$ & $(\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg (P\ \wedge \  Q)$
  & {\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[hilb19:3]{$3$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Negation of a disjunction:




\begin{thm}[hilb20]
\hypertarget{hilb20}{}
\begin{displaymath}
\neg (P\ \vee \  Q)\ \rightarrow \  (\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{hilb20:1}
  $1$ & $\neg \neg P\ \rightarrow \  P$
  & {\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{hilb20:2}
  $2$ & $\neg \neg Q\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb20:1]{$1$}
} \\
\label{hilb20:3}
  $3$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb20:4}
  $4$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb20:3]{$3$}
} \\
\label{hilb20:5}
  $5$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb20:4]{$4$}
} \\
\label{hilb20:6}
  $6$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb20:5]{$5$}
} \\
\label{hilb20:7}
  $7$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb20:6]{$6$}
} \\
\label{hilb20:8}
  $8$ & $(D\ \rightarrow \  P)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb20:7]{$7$}
} \\
\label{hilb20:9}
  $9$ & $(\neg \neg P\ \rightarrow \  P)\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg P$ in \hyperref[hilb20:8]{$8$}
} \\
\label{hilb20:10}
  $10$ & $(Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:1]{$1$}, \hyperref[hilb20:9]{$9$}} \\
\label{hilb20:11}
  $11$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb20:12}
  $12$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb20:11]{$11$}
} \\
\label{hilb20:13}
  $13$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb20:12]{$12$}
} \\
\label{hilb20:14}
  $14$ & $(B\ \vee \  P)\ \rightarrow \  (P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb20:13]{$13$}
} \\
\label{hilb20:15}
  $15$ & $(Q\ \vee \  P)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb20:14]{$14$}
} \\
\label{hilb20:16}
  $16$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb20:17}
  $17$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb20:16]{$16$}
} \\
\label{hilb20:18}
  $18$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb20:17]{$17$}
} \\
\label{hilb20:19}
  $19$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb20:18]{$18$}
} \\
\label{hilb20:20}
  $20$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \vee \  \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  \neg \neg P$ in \hyperref[hilb20:19]{$19$}
} \\
\label{hilb20:21}
  $21$ & $(D\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb20:20]{$20$}
} \\
\label{hilb20:22}
  $22$ & $((Q\ \vee \  P)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  P$ in \hyperref[hilb20:21]{$21$}
} \\
\label{hilb20:23}
  $23$ & $((Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:15]{$15$}, \hyperref[hilb20:22]{$22$}} \\
\label{hilb20:24}
  $24$ & $(Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:10]{$10$}, \hyperref[hilb20:23]{$23$}} \\
\label{hilb20:25}
  $25$ & $(B\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb20:13]{$13$}
} \\
\label{hilb20:26}
  $26$ & $(\neg \neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P$ in \hyperref[hilb20:25]{$25$}
} \\
\label{hilb20:27}
  $27$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb20:19]{$19$}
} \\
\label{hilb20:28}
  $28$ & $(D\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((\neg \neg P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb20:27]{$27$}
} \\
\label{hilb20:29}
  $29$ & $((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((\neg \neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg \neg P$ in \hyperref[hilb20:28]{$28$}
} \\
\label{hilb20:30}
  $30$ & $((\neg \neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:24]{$24$}, \hyperref[hilb20:29]{$29$}} \\
\label{hilb20:31}
  $31$ & $(\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:26]{$26$}, \hyperref[hilb20:30]{$30$}} \\
\label{hilb20:32}
  $32$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb20:33}
  $33$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb20:32]{$32$}
} \\
\label{hilb20:34}
  $34$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb20:33]{$33$}
} \\
\label{hilb20:35}
  $35$ & $(B\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb20:34]{$34$}
} \\
\label{hilb20:36}
  $36$ & $((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb20:35]{$35$}
} \\
\label{hilb20:37}
  $37$ & $\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:31]{$31$}, \hyperref[hilb20:36]{$36$}} \\
\label{hilb20:38}
  $38$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb20:39}
  $39$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb20:40}
  $40$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb20:38]{$38$}
} \\
\label{hilb20:41}
  $41$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb20:40]{$40$}
} \\
\label{hilb20:42}
  $42$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb20:39]{$39$}
} \\
\label{hilb20:43}
  $43$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb20:42]{$42$}
} \\
\label{hilb20:44}
  $44$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P$ in \hyperref[hilb20:6]{$6$}
} \\
\label{hilb20:45}
  $45$ & $(D\ \rightarrow \  Q)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q$ in \hyperref[hilb20:44]{$44$}
} \\
\label{hilb20:46}
  $46$ & $(\neg \neg Q\ \rightarrow \  Q)\ \rightarrow \  ((\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg Q$ in \hyperref[hilb20:45]{$45$}
} \\
\label{hilb20:47}
  $47$ & $(\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:2]{$2$}, \hyperref[hilb20:46]{$46$}} \\
\label{hilb20:48}
  $48$ & $(B\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb20:34]{$34$}
} \\
\label{hilb20:49}
  $49$ & $((\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  \neg \neg Q$ in \hyperref[hilb20:48]{$48$}
} \\
\label{hilb20:50}
  $50$ & $\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:47]{$47$}, \hyperref[hilb20:49]{$49$}} \\
\label{hilb20:51}
  $51$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb20:6]{$6$}
} \\
\label{hilb20:52}
  $52$ & $(D\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb20:51]{$51$}
} \\
\label{hilb20:53}
  $53$ & $(\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb20:52]{$52$}
} \\
\label{hilb20:54}
  $54$ & $(\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:50]{$50$}, \hyperref[hilb20:53]{$53$}} \\
\label{hilb20:55}
  $55$ & $(B\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg B\ \vee \  \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb20:41]{$41$}
} \\
\label{hilb20:56}
  $56$ & $(\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb20:55]{$55$}
} \\
\label{hilb20:57}
  $57$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb20:19]{$19$}
} \\
\label{hilb20:58}
  $58$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb20:57]{$57$}
} \\
\label{hilb20:59}
  $59$ & $((\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb20:58]{$58$}
} \\
\label{hilb20:60}
  $60$ & $((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:54]{$54$}, \hyperref[hilb20:59]{$59$}} \\
\label{hilb20:61}
  $61$ & $(\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:56]{$56$}, \hyperref[hilb20:60]{$60$}} \\
\label{hilb20:62}
  $62$ & $(\neg B\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (B\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb20:43]{$43$}
} \\
\label{hilb20:63}
  $63$ & $(\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb20:62]{$62$}
} \\
\label{hilb20:64}
  $64$ & $(D\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb20:57]{$57$}
} \\
\label{hilb20:65}
  $65$ & $((\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb20:64]{$64$}
} \\
\label{hilb20:66}
  $66$ & $((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:63]{$63$}, \hyperref[hilb20:65]{$65$}} \\
\label{hilb20:67}
  $67$ & $(\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:61]{$61$}, \hyperref[hilb20:66]{$66$}} \\
\label{hilb20:68}
  $68$ & $\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb20:37]{$37$}, \hyperref[hilb20:67]{$67$}} \\
\label{hilb20:69}
  $69$ & $\neg (P\ \vee \  Q)\ \rightarrow \  (\neg P\ \wedge \  \neg Q)$
  & {\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[hilb20:68]{$68$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Reverse of a negation of a disjunction:




\begin{thm}[hilb21]
\hypertarget{hilb21}{}
\begin{displaymath}
(\neg P\ \wedge \  \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb21:1}
  $1$ & $P\ \rightarrow \  \neg \neg P$
  & {\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{hilb21:2}
  $2$ & $Q\ \rightarrow \  \neg \neg Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb21:1]{$1$}
} \\
\label{hilb21:3}
  $3$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb21:4}
  $4$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb21:3]{$3$}
} \\
\label{hilb21:5}
  $5$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb21:4]{$4$}
} \\
\label{hilb21:6}
  $6$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb21:5]{$5$}
} \\
\label{hilb21:7}
  $7$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb21:6]{$6$}
} \\
\label{hilb21:8}
  $8$ & $(D\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  \neg \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg P$ in \hyperref[hilb21:7]{$7$}
} \\
\label{hilb21:9}
  $9$ & $(P\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P$ in \hyperref[hilb21:8]{$8$}
} \\
\label{hilb21:10}
  $10$ & $(Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:1]{$1$}, \hyperref[hilb21:9]{$9$}} \\
\label{hilb21:11}
  $11$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb21:12}
  $12$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb21:11]{$11$}
} \\
\label{hilb21:13}
  $13$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb21:12]{$12$}
} \\
\label{hilb21:14}
  $14$ & $(B\ \vee \  \neg \neg P)\ \rightarrow \  (\neg \neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P$ in \hyperref[hilb21:13]{$13$}
} \\
\label{hilb21:15}
  $15$ & $(Q\ \vee \  \neg \neg P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb21:14]{$14$}
} \\
\label{hilb21:16}
  $16$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb21:17}
  $17$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb21:16]{$16$}
} \\
\label{hilb21:18}
  $18$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb21:17]{$17$}
} \\
\label{hilb21:19}
  $19$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb21:18]{$18$}
} \\
\label{hilb21:20}
  $20$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \vee \  P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  P$ in \hyperref[hilb21:19]{$19$}
} \\
\label{hilb21:21}
  $21$ & $(D\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb21:20]{$20$}
} \\
\label{hilb21:22}
  $22$ & $((Q\ \vee \  \neg \neg P)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg \neg P$ in \hyperref[hilb21:21]{$21$}
} \\
\label{hilb21:23}
  $23$ & $((Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:15]{$15$}, \hyperref[hilb21:22]{$22$}} \\
\label{hilb21:24}
  $24$ & $(Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:10]{$10$}, \hyperref[hilb21:23]{$23$}} \\
\label{hilb21:25}
  $25$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb21:19]{$19$}
} \\
\label{hilb21:26}
  $26$ & $(D\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb21:25]{$25$}
} \\
\label{hilb21:27}
  $27$ & $((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  P$ in \hyperref[hilb21:26]{$26$}
} \\
\label{hilb21:28}
  $28$ & $((P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:24]{$24$}, \hyperref[hilb21:27]{$27$}} \\
\label{hilb21:29}
  $29$ & $(P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:11]{$11$}, \hyperref[hilb21:28]{$28$}} \\
\label{hilb21:30}
  $30$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb21:31}
  $31$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb21:30]{$30$}
} \\
\label{hilb21:32}
  $32$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb21:31]{$31$}
} \\
\label{hilb21:33}
  $33$ & $(B\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb21:32]{$32$}
} \\
\label{hilb21:34}
  $34$ & $((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb21:33]{$33$}
} \\
\label{hilb21:35}
  $35$ & $\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:29]{$29$}, \hyperref[hilb21:34]{$34$}} \\
\label{hilb21:36}
  $36$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb21:37}
  $37$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb21:38}
  $38$ & $(B\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb21:13]{$13$}
} \\
\label{hilb21:39}
  $39$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb21:36]{$36$}
} \\
\label{hilb21:40}
  $40$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb21:39]{$39$}
} \\
\label{hilb21:41}
  $41$ & $(B\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg B\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb21:40]{$40$}
} \\
\label{hilb21:42}
  $42$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb21:37]{$37$}
} \\
\label{hilb21:43}
  $43$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb21:42]{$42$}
} \\
\label{hilb21:44}
  $44$ & $(\neg B\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (B\ \rightarrow \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb21:43]{$43$}
} \\
\label{hilb21:45}
  $45$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P$ in \hyperref[hilb21:6]{$6$}
} \\
\label{hilb21:46}
  $46$ & $(D\ \rightarrow \  \neg \neg Q)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg Q$ in \hyperref[hilb21:45]{$45$}
} \\
\label{hilb21:47}
  $47$ & $(Q\ \rightarrow \  \neg \neg Q)\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q$ in \hyperref[hilb21:46]{$46$}
} \\
\label{hilb21:48}
  $48$ & $(\neg \neg P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:2]{$2$}, \hyperref[hilb21:47]{$47$}} \\
\label{hilb21:49}
  $49$ & $(B\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P\ \vee \  \neg \neg Q$ in \hyperref[hilb21:32]{$32$}
} \\
\label{hilb21:50}
  $50$ & $((\neg \neg P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb21:49]{$49$}
} \\
\label{hilb21:51}
  $51$ & $\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:48]{$48$}, \hyperref[hilb21:50]{$50$}} \\
\label{hilb21:52}
  $52$ & $(B\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb21:32]{$32$}
} \\
\label{hilb21:53}
  $53$ & $(\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb21:52]{$52$}
} \\
\label{hilb21:54}
  $54$ & $\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:51]{$51$}, \hyperref[hilb21:53]{$53$}} \\
\label{hilb21:55}
  $55$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb21:6]{$6$}
} \\
\label{hilb21:56}
  $56$ & $(D\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\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[hilb21:55]{$55$}
} \\
\label{hilb21:57}
  $57$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb21:56]{$56$}
} \\
\label{hilb21:58}
  $58$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:54]{$54$}, \hyperref[hilb21:57]{$57$}} \\
\label{hilb21:59}
  $59$ & $(B\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb21:13]{$13$}
} \\
\label{hilb21:60}
  $60$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb21:59]{$59$}
} \\
\label{hilb21:61}
  $61$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb21:19]{$19$}
} \\
\label{hilb21:62}
  $62$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:61]{$61$}
} \\
\label{hilb21:63}
  $63$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb21:62]{$62$}
} \\
\label{hilb21:64}
  $64$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:60]{$60$}, \hyperref[hilb21:63]{$63$}} \\
\label{hilb21:65}
  $65$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:58]{$58$}, \hyperref[hilb21:64]{$64$}} \\
\label{hilb21:66}
  $66$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb21:38]{$38$}
} \\
\label{hilb21:67}
  $67$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:19]{$19$}
} \\
\label{hilb21:68}
  $68$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:67]{$67$}
} \\
\label{hilb21:69}
  $69$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))))$
  & {\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[hilb21:68]{$68$}
} \\
\label{hilb21:70}
  $70$ & $((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:65]{$65$}, \hyperref[hilb21:69]{$69$}} \\
\label{hilb21:71}
  $71$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:66]{$66$}, \hyperref[hilb21:70]{$70$}} \\
\label{hilb21:72}
  $72$ & $(\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb21:41]{$41$}
} \\
\label{hilb21:73}
  $73$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:19]{$19$}
} \\
\label{hilb21:74}
  $74$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:73]{$73$}
} \\
\label{hilb21:75}
  $75$ & $((\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:74]{$74$}
} \\
\label{hilb21:76}
  $76$ & $((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:71]{$71$}, \hyperref[hilb21:75]{$75$}} \\
\label{hilb21:77}
  $77$ & $(\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:72]{$72$}, \hyperref[hilb21:76]{$76$}} \\
\label{hilb21:78}
  $78$ & $(\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb21:44]{$44$}
} \\
\label{hilb21:79}
  $79$ & $(D\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:73]{$73$}
} \\
\label{hilb21:80}
  $80$ & $((\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q)))\ \rightarrow \  (((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb21:79]{$79$}
} \\
\label{hilb21:81}
  $81$ & $((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:78]{$78$}, \hyperref[hilb21:80]{$80$}} \\
\label{hilb21:82}
  $82$ & $(\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:77]{$77$}, \hyperref[hilb21:81]{$81$}} \\
\label{hilb21:83}
  $83$ & $\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb21:35]{$35$}, \hyperref[hilb21:82]{$82$}} \\
\label{hilb21:84}
  $84$ & $(\neg P\ \wedge \  \neg Q)\ \rightarrow \  \neg (P\ \vee \  Q)$
  & {\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[hilb21:83]{$83$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



The Conjunction is commutative:




\begin{thm}[hilb22]
\hypertarget{hilb22}{}
\begin{displaymath}
(P\ \wedge \  Q)\ \rightarrow \  (Q\ \wedge \  P)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb22:1}
  $1$ & $P\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb2}{hilb2} } \\
\label{hilb22:2}
  $2$ & $Q\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb22:1]{$1$}
} \\
\label{hilb22:3}
  $3$ & $(P\ \wedge \  Q)\ \rightarrow \  (P\ \wedge \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $P\ \wedge \  Q$ in \hyperref[hilb22:2]{$2$}
} \\
\label{hilb22:4}
  $4$ & $(P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule5}{use abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb22:3]{$3$} at occurence $2$
} \\
\label{hilb22:5}
  $5$ & $(Q\ \vee \  P)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb10}{hilb10} } \\
\label{hilb22:6}
  $6$ & $(A\ \vee \  P)\ \rightarrow \  (P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb22:5]{$5$}
} \\
\label{hilb22:7}
  $7$ & $(A\ \vee \  B)\ \rightarrow \  (B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb22:6]{$6$}
} \\
\label{hilb22:8}
  $8$ & $(\neg Q\ \vee \  B)\ \rightarrow \  (B\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg Q$ in \hyperref[hilb22:7]{$7$}
} \\
\label{hilb22:9}
  $9$ & $(\neg Q\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P$ in \hyperref[hilb22:8]{$8$}
} \\
\label{hilb22:10}
  $10$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb22:11}
  $11$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb22:10]{$10$}
} \\
\label{hilb22:12}
  $12$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb22:11]{$11$}
} \\
\label{hilb22:13}
  $13$ & $(B\ \rightarrow \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb22:12]{$12$}
} \\
\label{hilb22:14}
  $14$ & $((\neg Q\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg Q\ \vee \  \neg P$ in \hyperref[hilb22:13]{$13$}
} \\
\label{hilb22:15}
  $15$ & $\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb22:9]{$9$}, \hyperref[hilb22:14]{$14$}} \\
\label{hilb22:16}
  $16$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb22:17}
  $17$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb22:18}
  $18$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb22:19}
  $19$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb22:18]{$18$}
} \\
\label{hilb22:20}
  $20$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb22:19]{$19$}
} \\
\label{hilb22:21}
  $21$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb22:20]{$20$}
} \\
\label{hilb22:22}
  $22$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \wedge \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \wedge \  Q)$ in \hyperref[hilb22:21]{$21$}
} \\
\label{hilb22:23}
  $23$ & $(D\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))\ \rightarrow \  ((\neg (P\ \wedge \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg Q\ \vee \  \neg P)$ in \hyperref[hilb22:22]{$22$}
} \\
\label{hilb22:24}
  $24$ & $(\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))\ \rightarrow \  ((\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb22:23]{$23$}
} \\
\label{hilb22:25}
  $25$ & $(\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb22:15]{$15$}, \hyperref[hilb22:24]{$24$}} \\
\label{hilb22:26}
  $26$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb22:16]{$16$}
} \\
\label{hilb22:27}
  $27$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb22:26]{$26$}
} \\
\label{hilb22:28}
  $28$ & $(B\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg B\ \vee \  \neg (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb22:27]{$27$}
} \\
\label{hilb22:29}
  $29$ & $((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  Q$ in \hyperref[hilb22:28]{$28$}
} \\
\label{hilb22:30}
  $30$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb22:31}
  $31$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb22:30]{$30$}
} \\
\label{hilb22:32}
  $32$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb22:31]{$31$}
} \\
\label{hilb22:33}
  $33$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb22:32]{$32$}
} \\
\label{hilb22:34}
  $34$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb22:33]{$33$}
} \\
\label{hilb22:35}
  $35$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)$ in \hyperref[hilb22:34]{$34$}
} \\
\label{hilb22:36}
  $36$ & $((\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q)))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb22:35]{$35$}
} \\
\label{hilb22:37}
  $37$ & $(((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q)))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb22:25]{$25$}, \hyperref[hilb22:36]{$36$}} \\
\label{hilb22:38}
  $38$ & $((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb22:29]{$29$}, \hyperref[hilb22:37]{$37$}} \\
\label{hilb22:39}
  $39$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb22:17]{$17$}
} \\
\label{hilb22:40}
  $40$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb22:39]{$39$}
} \\
\label{hilb22:41}
  $41$ & $(\neg B\ \vee \  \neg (\neg Q\ \vee \  \neg P))\ \rightarrow \  (B\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg Q\ \vee \  \neg P)$ in \hyperref[hilb22:40]{$40$}
} \\
\label{hilb22:42}
  $42$ & $(\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  Q$ in \hyperref[hilb22:41]{$41$}
} \\
\label{hilb22:43}
  $43$ & $(D\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P)))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P)$ in \hyperref[hilb22:34]{$34$}
} \\
\label{hilb22:44}
  $44$ & $((\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P)))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)$ in \hyperref[hilb22:43]{$43$}
} \\
\label{hilb22:45}
  $45$ & $(((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg (\neg Q\ \vee \  \neg P)))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb22:42]{$42$}, \hyperref[hilb22:44]{$44$}} \\
\label{hilb22:46}
  $46$ & $((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb22:38]{$38$}, \hyperref[hilb22:45]{$45$}} \\
\label{hilb22:47}
  $47$ & $(P\ \wedge \  Q)\ \rightarrow \  \neg (\neg Q\ \vee \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb22:4]{$4$}, \hyperref[hilb22:46]{$46$}} \\
\label{hilb22:48}
  $48$ & $(P\ \wedge \  Q)\ \rightarrow \  (Q\ \wedge \  P)$
  & {\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[hilb22:47]{$47$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



A technical lemma that is simular to the previous one:




\begin{thm}[hilb23]
\hypertarget{hilb23}{}
\begin{displaymath}
(Q\ \wedge \  P)\ \rightarrow \  (P\ \wedge \  Q)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb23:1}
  $1$ & $(P\ \wedge \  Q)\ \rightarrow \  (Q\ \wedge \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb22}{hilb22} } \\
\label{hilb23:2}
  $2$ & $(P\ \wedge \  A)\ \rightarrow \  (A\ \wedge \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb23:1]{$1$}
} \\
\label{hilb23:3}
  $3$ & $(B\ \wedge \  A)\ \rightarrow \  (A\ \wedge \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb23:2]{$2$}
} \\
\label{hilb23:4}
  $4$ & $(B\ \wedge \  P)\ \rightarrow \  (P\ \wedge \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb23:3]{$3$}
} \\
\label{hilb23:5}
  $5$ & $(Q\ \wedge \  P)\ \rightarrow \  (P\ \wedge \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb23:4]{$4$}
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Reduction of a conjunction:




\begin{thm}[hilb24]
\hypertarget{hilb24}{}
\begin{displaymath}
(P\ \wedge \  Q)\ \rightarrow \  P\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb24:1}
  $1$ & $P\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom2}{axiom2} } \\
\label{hilb24:2}
  $2$ & $P\ \rightarrow \  (P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb24:1]{$1$}
} \\
\label{hilb24:3}
  $3$ & $B\ \rightarrow \  (B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb24:2]{$2$}
} \\
\label{hilb24:4}
  $4$ & $B\ \rightarrow \  (B\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg Q$ in \hyperref[hilb24:3]{$3$}
} \\
\label{hilb24:5}
  $5$ & $\neg P\ \rightarrow \  (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P$ in \hyperref[hilb24:4]{$4$}
} \\
\label{hilb24:6}
  $6$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb24:7}
  $7$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb24:6]{$6$}
} \\
\label{hilb24:8}
  $8$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb24:7]{$7$}
} \\
\label{hilb24:9}
  $9$ & $(B\ \rightarrow \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb24:8]{$8$}
} \\
\label{hilb24:10}
  $10$ & $(\neg P\ \rightarrow \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P$ in \hyperref[hilb24:9]{$9$}
} \\
\label{hilb24:11}
  $11$ & $\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg \neg P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb24:5]{$5$}, \hyperref[hilb24:10]{$10$}} \\
\label{hilb24:12}
  $12$ & $(P\ \wedge \  Q)\ \rightarrow \  \neg \neg P$
  & {\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[hilb24:11]{$11$} at occurence $1$
} \\
\label{hilb24:13}
  $13$ & $\neg \neg P\ \rightarrow \  P$
  & {\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{hilb24:14}
  $14$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb24:15}
  $15$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb24:16}
  $16$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb24:17}
  $17$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb24:16]{$16$}
} \\
\label{hilb24:18}
  $18$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb24:17]{$17$}
} \\
\label{hilb24:19}
  $19$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb24:18]{$18$}
} \\
\label{hilb24:20}
  $20$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \wedge \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \wedge \  Q)$ in \hyperref[hilb24:19]{$19$}
} \\
\label{hilb24:21}
  $21$ & $(D\ \rightarrow \  P)\ \rightarrow \  ((\neg (P\ \wedge \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb24:20]{$20$}
} \\
\label{hilb24:22}
  $22$ & $(\neg \neg P\ \rightarrow \  P)\ \rightarrow \  ((\neg (P\ \wedge \  Q)\ \vee \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg P$ in \hyperref[hilb24:21]{$21$}
} \\
\label{hilb24:23}
  $23$ & $(\neg (P\ \wedge \  Q)\ \vee \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb24:13]{$13$}, \hyperref[hilb24:22]{$22$}} \\
\label{hilb24:24}
  $24$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb24:14]{$14$}
} \\
\label{hilb24:25}
  $25$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb24:24]{$24$}
} \\
\label{hilb24:26}
  $26$ & $(B\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg B\ \vee \  \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P$ in \hyperref[hilb24:25]{$25$}
} \\
\label{hilb24:27}
  $27$ & $((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  Q$ in \hyperref[hilb24:26]{$26$}
} \\
\label{hilb24:28}
  $28$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb24:29}
  $29$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb24:28]{$28$}
} \\
\label{hilb24:30}
  $30$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb24:29]{$29$}
} \\
\label{hilb24:31}
  $31$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb24:30]{$30$}
} \\
\label{hilb24:32}
  $32$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \wedge \  Q)\ \rightarrow \  \neg \neg P$ in \hyperref[hilb24:31]{$31$}
} \\
\label{hilb24:33}
  $33$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  Q)\ \vee \  P$ in \hyperref[hilb24:32]{$32$}
} \\
\label{hilb24:34}
  $34$ & $((\neg (P\ \wedge \  Q)\ \vee \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg \neg P))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  Q)\ \vee \  \neg \neg P$ in \hyperref[hilb24:33]{$33$}
} \\
\label{hilb24:35}
  $35$ & $(((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  \neg \neg P))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb24:23]{$23$}, \hyperref[hilb24:34]{$34$}} \\
\label{hilb24:36}
  $36$ & $((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb24:27]{$27$}, \hyperref[hilb24:35]{$35$}} \\
\label{hilb24:37}
  $37$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb24:15]{$15$}
} \\
\label{hilb24:38}
  $38$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb24:37]{$37$}
} \\
\label{hilb24:39}
  $39$ & $(\neg B\ \vee \  P)\ \rightarrow \  (B\ \rightarrow \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb24:38]{$38$}
} \\
\label{hilb24:40}
  $40$ & $(\neg (P\ \wedge \  Q)\ \vee \  P)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  Q$ in \hyperref[hilb24:39]{$39$}
} \\
\label{hilb24:41}
  $41$ & $(D\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  P))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  Q)\ \rightarrow \  P$ in \hyperref[hilb24:32]{$32$}
} \\
\label{hilb24:42}
  $42$ & $((\neg (P\ \wedge \  Q)\ \vee \  P)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  P))\ \rightarrow \  ((((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  Q)\ \vee \  P$ in \hyperref[hilb24:41]{$41$}
} \\
\label{hilb24:43}
  $43$ & $(((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  P))\ \rightarrow \  (((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb24:40]{$40$}, \hyperref[hilb24:42]{$42$}} \\
\label{hilb24:44}
  $44$ & $((P\ \wedge \  Q)\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb24:36]{$36$}, \hyperref[hilb24:43]{$43$}} \\
\label{hilb24:45}
  $45$ & $(P\ \wedge \  Q)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb24:12]{$12$}, \hyperref[hilb24:44]{$44$}} \\
 & & \qedhere
\end{longtable}
\end{proof}



Another form of a reduction of a conjunction:




\begin{thm}[hilb25]
\hypertarget{hilb25}{}
\begin{displaymath}
(P\ \wedge \  Q)\ \rightarrow \  Q\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb25:1}
  $1$ & $(P\ \wedge \  Q)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb24}{hilb24} } \\
\label{hilb25:2}
  $2$ & $(P\ \wedge \  A)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb25:1]{$1$}
} \\
\label{hilb25:3}
  $3$ & $(B\ \wedge \  A)\ \rightarrow \  B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb25:2]{$2$}
} \\
\label{hilb25:4}
  $4$ & $(B\ \wedge \  P)\ \rightarrow \  B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb25:3]{$3$}
} \\
\label{hilb25:5}
  $5$ & $(Q\ \wedge \  P)\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb25:4]{$4$}
} \\
\label{hilb25:6}
  $6$ & $(Q\ \wedge \  P)\ \rightarrow \  (P\ \wedge \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb23}{hilb23} } \\
\label{hilb25:7}
  $7$ & $(A\ \wedge \  P)\ \rightarrow \  (P\ \wedge \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb25:6]{$6$}
} \\
\label{hilb25:8}
  $8$ & $(A\ \wedge \  B)\ \rightarrow \  (B\ \wedge \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb25:7]{$7$}
} \\
\label{hilb25:9}
  $9$ & $(P\ \wedge \  B)\ \rightarrow \  (B\ \wedge \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb25:8]{$8$}
} \\
\label{hilb25:10}
  $10$ & $(P\ \wedge \  Q)\ \rightarrow \  (Q\ \wedge \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb25:9]{$9$}
} \\
\label{hilb25:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb25:12}
  $12$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb25:13}
  $13$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb25:14}
  $14$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb25:13]{$13$}
} \\
\label{hilb25:15}
  $15$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb25:14]{$14$}
} \\
\label{hilb25:16}
  $16$ & $(B\ \rightarrow \  (Q\ \wedge \  P))\ \rightarrow \  (\neg (Q\ \wedge \  P)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q\ \wedge \  P$ in \hyperref[hilb25:15]{$15$}
} \\
\label{hilb25:17}
  $17$ & $((P\ \wedge \  Q)\ \rightarrow \  (Q\ \wedge \  P))\ \rightarrow \  (\neg (Q\ \wedge \  P)\ \rightarrow \  \neg (P\ \wedge \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  Q$ in \hyperref[hilb25:16]{$16$}
} \\
\label{hilb25:18}
  $18$ & $\neg (Q\ \wedge \  P)\ \rightarrow \  \neg (P\ \wedge \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:10]{$10$}, \hyperref[hilb25:17]{$17$}} \\
\label{hilb25:19}
  $19$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb25:20}
  $20$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb25:19]{$19$}
} \\
\label{hilb25:21}
  $21$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb25:20]{$20$}
} \\
\label{hilb25:22}
  $22$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb25:21]{$21$}
} \\
\label{hilb25:23}
  $23$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb25:22]{$22$}
} \\
\label{hilb25:24}
  $24$ & $(D\ \rightarrow \  \neg (P\ \wedge \  Q))\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  Q)$ in \hyperref[hilb25:23]{$23$}
} \\
\label{hilb25:25}
  $25$ & $(\neg (Q\ \wedge \  P)\ \rightarrow \  \neg (P\ \wedge \  Q))\ \rightarrow \  ((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (Q\ \wedge \  P)$ in \hyperref[hilb25:24]{$24$}
} \\
\label{hilb25:26}
  $26$ & $(Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:18]{$18$}, \hyperref[hilb25:25]{$25$}} \\
\label{hilb25:27}
  $27$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb25:28}
  $28$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb25:27]{$27$}
} \\
\label{hilb25:29}
  $29$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb25:28]{$28$}
} \\
\label{hilb25:30}
  $30$ & $(B\ \vee \  \neg (P\ \wedge \  Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (P\ \wedge \  Q)$ in \hyperref[hilb25:29]{$29$}
} \\
\label{hilb25:31}
  $31$ & $(Q\ \vee \  \neg (P\ \wedge \  Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb25:30]{$30$}
} \\
\label{hilb25:32}
  $32$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb25:33}
  $33$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb25:32]{$32$}
} \\
\label{hilb25:34}
  $34$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb25:33]{$33$}
} \\
\label{hilb25:35}
  $35$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb25:34]{$34$}
} \\
\label{hilb25:36}
  $36$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  \neg (Q\ \wedge \  P)$ in \hyperref[hilb25:35]{$35$}
} \\
\label{hilb25:37}
  $37$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  Q)\ \vee \  Q$ in \hyperref[hilb25:36]{$36$}
} \\
\label{hilb25:38}
  $38$ & $((Q\ \vee \  \neg (P\ \wedge \  Q))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  Q)))\ \rightarrow \  ((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg (P\ \wedge \  Q)$ in \hyperref[hilb25:37]{$37$}
} \\
\label{hilb25:39}
  $39$ & $((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  Q)))\ \rightarrow \  ((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:31]{$31$}, \hyperref[hilb25:38]{$38$}} \\
\label{hilb25:40}
  $40$ & $(Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:26]{$26$}, \hyperref[hilb25:39]{$39$}} \\
\label{hilb25:41}
  $41$ & $(B\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb25:29]{$29$}
} \\
\label{hilb25:42}
  $42$ & $(\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg (Q\ \wedge \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (Q\ \wedge \  P)$ in \hyperref[hilb25:41]{$41$}
} \\
\label{hilb25:43}
  $43$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (Q\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb25:35]{$35$}
} \\
\label{hilb25:44}
  $44$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  (((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  Q)\ \vee \  Q$ in \hyperref[hilb25:43]{$43$}
} \\
\label{hilb25:45}
  $45$ & $((Q\ \vee \  \neg (Q\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  (((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg (Q\ \wedge \  P)))\ \rightarrow \  ((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg (Q\ \wedge \  P)$ in \hyperref[hilb25:44]{$44$}
} \\
\label{hilb25:46}
  $46$ & $((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg (Q\ \wedge \  P)))\ \rightarrow \  ((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:40]{$40$}, \hyperref[hilb25:45]{$45$}} \\
\label{hilb25:47}
  $47$ & $(\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:42]{$42$}, \hyperref[hilb25:46]{$46$}} \\
\label{hilb25:48}
  $48$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb25:11]{$11$}
} \\
\label{hilb25:49}
  $49$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb25:48]{$48$}
} \\
\label{hilb25:50}
  $50$ & $(B\ \rightarrow \  Q)\ \rightarrow \  (\neg B\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb25:49]{$49$}
} \\
\label{hilb25:51}
  $51$ & $((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (Q\ \wedge \  P)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \wedge \  P$ in \hyperref[hilb25:50]{$50$}
} \\
\label{hilb25:52}
  $52$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  (((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(Q\ \wedge \  P)\ \rightarrow \  Q$ in \hyperref[hilb25:35]{$35$}
} \\
\label{hilb25:53}
  $53$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  ((((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  (((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  Q)\ \vee \  Q$ in \hyperref[hilb25:52]{$52$}
} \\
\label{hilb25:54}
  $54$ & $((\neg (Q\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  ((((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (Q\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (Q\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb25:53]{$53$}
} \\
\label{hilb25:55}
  $55$ & $(((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (Q\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:47]{$47$}, \hyperref[hilb25:54]{$54$}} \\
\label{hilb25:56}
  $56$ & $((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:51]{$51$}, \hyperref[hilb25:55]{$55$}} \\
\label{hilb25:57}
  $57$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb25:12]{$12$}
} \\
\label{hilb25:58}
  $58$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb25:57]{$57$}
} \\
\label{hilb25:59}
  $59$ & $(\neg B\ \vee \  Q)\ \rightarrow \  (B\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb25:58]{$58$}
} \\
\label{hilb25:60}
  $60$ & $(\neg (P\ \wedge \  Q)\ \vee \  Q)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  Q$ in \hyperref[hilb25:59]{$59$}
} \\
\label{hilb25:61}
  $61$ & $(D\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  Q))\ \rightarrow \  ((((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  (((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  Q)\ \rightarrow \  Q$ in \hyperref[hilb25:52]{$52$}
} \\
\label{hilb25:62}
  $62$ & $((\neg (P\ \wedge \  Q)\ \vee \  Q)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  Q))\ \rightarrow \  ((((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  (((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  Q)\ \vee \  Q$ in \hyperref[hilb25:61]{$61$}
} \\
\label{hilb25:63}
  $63$ & $(((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  Q)\ \vee \  Q))\ \rightarrow \  (((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:60]{$60$}, \hyperref[hilb25:62]{$62$}} \\
\label{hilb25:64}
  $64$ & $((Q\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:56]{$56$}, \hyperref[hilb25:63]{$63$}} \\
\label{hilb25:65}
  $65$ & $(P\ \wedge \  Q)\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb25:5]{$5$}, \hyperref[hilb25:64]{$64$}} \\
 & & \qedhere
\end{longtable}
\end{proof}



The conjunction is associative too (first implication):




\begin{thm}[hilb26]
\hypertarget{hilb26}{}
\begin{displaymath}
(P\ \wedge \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \wedge \  A)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb26:1}
  $1$ & $((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb15}{hilb15} } \\
\label{hilb26:2}
  $2$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb26:3}
  $3$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb26:2]{$2$}
} \\
\label{hilb26:4}
  $4$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb26:3]{$3$}
} \\
\label{hilb26:5}
  $5$ & $(B\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  (Q\ \vee \  A)$ in \hyperref[hilb26:4]{$4$}
} \\
\label{hilb26:6}
  $6$ & $(((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb26:5]{$5$}
} \\
\label{hilb26:7}
  $7$ & $\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:1]{$1$}, \hyperref[hilb26:6]{$6$}} \\
\label{hilb26:8}
  $8$ & $P\ \rightarrow \  \neg \neg P$
  & {\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{hilb26:9}
  $9$ & $B\ \rightarrow \  \neg \neg B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb26:8]{$8$}
} \\
\label{hilb26:10}
  $10$ & $(Q\ \vee \  A)\ \rightarrow \  \neg \neg (Q\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  A$ in \hyperref[hilb26:9]{$9$}
} \\
\label{hilb26:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb26:12}
  $12$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb26:11]{$11$}
} \\
\label{hilb26:13}
  $13$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb26:12]{$12$}
} \\
\label{hilb26:14}
  $14$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb26:13]{$13$}
} \\
\label{hilb26:15}
  $15$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb26:14]{$14$}
} \\
\label{hilb26:16}
  $16$ & $(D\ \rightarrow \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (Q\ \vee \  A)$ in \hyperref[hilb26:15]{$15$}
} \\
\label{hilb26:17}
  $17$ & $((Q\ \vee \  A)\ \rightarrow \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  A$ in \hyperref[hilb26:16]{$16$}
} \\
\label{hilb26:18}
  $18$ & $(P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:10]{$10$}, \hyperref[hilb26:17]{$17$}} \\
\label{hilb26:19}
  $19$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb26:2]{$2$}
} \\
\label{hilb26:20}
  $20$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb26:19]{$19$}
} \\
\label{hilb26:21}
  $21$ & $(C\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  \neg \neg (Q\ \vee \  A)$ in \hyperref[hilb26:20]{$20$}
} \\
\label{hilb26:22}
  $22$ & $((P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  (Q\ \vee \  A)$ in \hyperref[hilb26:21]{$21$}
} \\
\label{hilb26:23}
  $23$ & $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:18]{$18$}, \hyperref[hilb26:22]{$22$}} \\
\label{hilb26:24}
  $24$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb26:25}
  $25$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb26:26}
  $26$ & $(C\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb26:20]{$20$}
} \\
\label{hilb26:27}
  $27$ & $(\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:26]{$26$}
} \\
\label{hilb26:28}
  $28$ & $\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:23]{$23$}, \hyperref[hilb26:27]{$27$}} \\
\label{hilb26:29}
  $29$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:14]{$14$}
} \\
\label{hilb26:30}
  $30$ & $(D\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:29]{$29$}
} \\
\label{hilb26:31}
  $31$ & $(\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb26:30]{$30$}
} \\
\label{hilb26:32}
  $32$ & $(\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:28]{$28$}, \hyperref[hilb26:31]{$31$}} \\
\label{hilb26:33}
  $33$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb26:34}
  $34$ & $(P\ \vee \  B)\ \rightarrow \  (B\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb26:33]{$33$}
} \\
\label{hilb26:35}
  $35$ & $(C\ \vee \  B)\ \rightarrow \  (B\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb26:34]{$34$}
} \\
\label{hilb26:36}
  $36$ & $(C\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:35]{$35$}
} \\
\label{hilb26:37}
  $37$ & $(\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:36]{$36$}
} \\
\label{hilb26:38}
  $38$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb26:39}
  $39$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb26:38]{$38$}
} \\
\label{hilb26:40}
  $40$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb26:39]{$39$}
} \\
\label{hilb26:41}
  $41$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb26:40]{$40$}
} \\
\label{hilb26:42}
  $42$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb26:41]{$41$}
} \\
\label{hilb26:43}
  $43$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:42]{$42$}
} \\
\label{hilb26:44}
  $44$ & $((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:43]{$43$}
} \\
\label{hilb26:45}
  $45$ & $((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:37]{$37$}, \hyperref[hilb26:44]{$44$}} \\
\label{hilb26:46}
  $46$ & $(\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:32]{$32$}, \hyperref[hilb26:45]{$45$}} \\
\label{hilb26:47}
  $47$ & $(C\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:35]{$35$}
} \\
\label{hilb26:48}
  $48$ & $(\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb26:47]{$47$}
} \\
\label{hilb26:49}
  $49$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:41]{$41$}
} \\
\label{hilb26:50}
  $50$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:49]{$49$}
} \\
\label{hilb26:51}
  $51$ & $((\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb26:50]{$50$}
} \\
\label{hilb26:52}
  $52$ & $((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:46]{$46$}, \hyperref[hilb26:51]{$51$}} \\
\label{hilb26:53}
  $53$ & $(\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:48]{$48$}, \hyperref[hilb26:52]{$52$}} \\
\label{hilb26:54}
  $54$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb26:24]{$24$}
} \\
\label{hilb26:55}
  $55$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb26:54]{$54$}
} \\
\label{hilb26:56}
  $56$ & $(C\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg C\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:55]{$55$}
} \\
\label{hilb26:57}
  $57$ & $(\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb26:56]{$56$}
} \\
\label{hilb26:58}
  $58$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:41]{$41$}
} \\
\label{hilb26:59}
  $59$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:58]{$58$}
} \\
\label{hilb26:60}
  $60$ & $((\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:59]{$59$}
} \\
\label{hilb26:61}
  $61$ & $((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:53]{$53$}, \hyperref[hilb26:60]{$60$}} \\
\label{hilb26:62}
  $62$ & $(\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:57]{$57$}, \hyperref[hilb26:61]{$61$}} \\
\label{hilb26:63}
  $63$ & $(\neg P\ \vee \  B)\ \rightarrow \  (P\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb26:25]{$25$}
} \\
\label{hilb26:64}
  $64$ & $(\neg C\ \vee \  B)\ \rightarrow \  (C\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb26:63]{$63$}
} \\
\label{hilb26:65}
  $65$ & $(\neg C\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (C\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:64]{$64$}
} \\
\label{hilb26:66}
  $66$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:65]{$65$}
} \\
\label{hilb26:67}
  $67$ & $(D\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:58]{$58$}
} \\
\label{hilb26:68}
  $68$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:67]{$67$}
} \\
\label{hilb26:69}
  $69$ & $((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:66]{$66$}, \hyperref[hilb26:68]{$68$}} \\
\label{hilb26:70}
  $70$ & $(\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:62]{$62$}, \hyperref[hilb26:69]{$69$}} \\
\label{hilb26:71}
  $71$ & $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:7]{$7$}, \hyperref[hilb26:70]{$70$}} \\
\label{hilb26:72}
  $72$ & $\neg \neg P\ \rightarrow \  P$
  & {\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{hilb26:73}
  $73$ & $\neg \neg A\ \rightarrow \  A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb26:72]{$72$}
} \\
\label{hilb26:74}
  $74$ & $\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb26:73]{$73$}
} \\
\label{hilb26:75}
  $75$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb26:14]{$14$}
} \\
\label{hilb26:76}
  $76$ & $(D\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb26:75]{$75$}
} \\
\label{hilb26:77}
  $77$ & $(\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  ((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb26:76]{$76$}
} \\
\label{hilb26:78}
  $78$ & $(A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:74]{$74$}, \hyperref[hilb26:77]{$77$}} \\
\label{hilb26:79}
  $79$ & $(C\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb26:35]{$35$}
} \\
\label{hilb26:80}
  $80$ & $(A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $A$ in \hyperref[hilb26:79]{$79$}
} \\
\label{hilb26:81}
  $81$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A\ \vee \  \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb26:41]{$41$}
} \\
\label{hilb26:82}
  $82$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb26:81]{$81$}
} \\
\label{hilb26:83}
  $83$ & $((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb26:82]{$82$}
} \\
\label{hilb26:84}
  $84$ & $((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:80]{$80$}, \hyperref[hilb26:83]{$83$}} \\
\label{hilb26:85}
  $85$ & $(A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:78]{$78$}, \hyperref[hilb26:84]{$84$}} \\
\label{hilb26:86}
  $86$ & $(C\ \vee \  A)\ \rightarrow \  (A\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb26:35]{$35$}
} \\
\label{hilb26:87}
  $87$ & $(\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb26:86]{$86$}
} \\
\label{hilb26:88}
  $88$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb26:41]{$41$}
} \\
\label{hilb26:89}
  $89$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb26:88]{$88$}
} \\
\label{hilb26:90}
  $90$ & $((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb26:89]{$89$}
} \\
\label{hilb26:91}
  $91$ & $((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:85]{$85$}, \hyperref[hilb26:90]{$90$}} \\
\label{hilb26:92}
  $92$ & $(\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:87]{$87$}, \hyperref[hilb26:91]{$91$}} \\
\label{hilb26:93}
  $93$ & $(C\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb26:20]{$20$}
} \\
\label{hilb26:94}
  $94$ & $((\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb26:93]{$93$}
} \\
\label{hilb26:95}
  $95$ & $\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:92]{$92$}, \hyperref[hilb26:94]{$94$}} \\
\label{hilb26:96}
  $96$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:14]{$14$}
} \\
\label{hilb26:97}
  $97$ & $(D\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:96]{$96$}
} \\
\label{hilb26:98}
  $98$ & $(\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:97]{$97$}
} \\
\label{hilb26:99}
  $99$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:95]{$95$}, \hyperref[hilb26:98]{$98$}} \\
\label{hilb26:100}
  $100$ & $(\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:56]{$56$}
} \\
\label{hilb26:101}
  $101$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:41]{$41$}
} \\
\label{hilb26:102}
  $102$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:101]{$101$}
} \\
\label{hilb26:103}
  $103$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:102]{$102$}
} \\
\label{hilb26:104}
  $104$ & $((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:99]{$99$}, \hyperref[hilb26:103]{$103$}} \\
\label{hilb26:105}
  $105$ & $(\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:100]{$100$}, \hyperref[hilb26:104]{$104$}} \\
\label{hilb26:106}
  $106$ & $(\neg C\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (C\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:64]{$64$}
} \\
\label{hilb26:107}
  $107$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb26:106]{$106$}
} \\
\label{hilb26:108}
  $108$ & $(D\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:101]{$101$}
} \\
\label{hilb26:109}
  $109$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb26:108]{$108$}
} \\
\label{hilb26:110}
  $110$ & $((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:107]{$107$}, \hyperref[hilb26:109]{$109$}} \\
\label{hilb26:111}
  $111$ & $(\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:105]{$105$}, \hyperref[hilb26:110]{$110$}} \\
\label{hilb26:112}
  $112$ & $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb26:71]{$71$}, \hyperref[hilb26:111]{$111$}} \\
\label{hilb26:113}
  $113$ & $\neg (P\ \vee \  \neg \neg (Q\ \vee \  B))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb26:112]{$112$}
} \\
\label{hilb26:114}
  $114$ & $\neg (P\ \vee \  \neg \neg (C\ \vee \  B))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  C)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb26:113]{$113$}
} \\
\label{hilb26:115}
  $115$ & $\neg (D\ \vee \  \neg \neg (C\ \vee \  B))\ \rightarrow \  \neg (\neg \neg (D\ \vee \  C)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb26:114]{$114$}
} \\
\label{hilb26:116}
  $116$ & $\neg (D\ \vee \  \neg \neg (C\ \vee \  \neg A))\ \rightarrow \  \neg (\neg \neg (D\ \vee \  C)\ \vee \  \neg A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg A$ in \hyperref[hilb26:115]{$115$}
} \\
\label{hilb26:117}
  $117$ & $\neg (D\ \vee \  \neg \neg (\neg Q\ \vee \  \neg A))\ \rightarrow \  \neg (\neg \neg (D\ \vee \  \neg Q)\ \vee \  \neg A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg Q$ in \hyperref[hilb26:116]{$116$}
} \\
\label{hilb26:118}
  $118$ & $\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \vee \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb26:117]{$117$}
} \\
\label{hilb26:119}
  $119$ & $(P\ \wedge \  \neg (\neg Q\ \vee \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg A)$
  & {\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[hilb26:118]{$118$} at occurence $1$
} \\
\label{hilb26:120}
  $120$ & $(P\ \wedge \  (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg A)$
  & {\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[hilb26:119]{$119$} at occurence $1$
} \\
\label{hilb26:121}
  $121$ & $(P\ \wedge \  (Q\ \wedge \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \wedge \  A)$
  & {\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[hilb26:120]{$120$} at occurence $1$
} \\
\label{hilb26:122}
  $122$ & $(P\ \wedge \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \wedge \  A)$
  & {\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[hilb26:121]{$121$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



The conjunction is associative (second implication):




\begin{thm}[hilb27]
\hypertarget{hilb27}{}
\begin{displaymath}
((P\ \wedge \  Q)\ \wedge \  A)\ \rightarrow \  (P\ \wedge \  (Q\ \wedge \  A))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb27:1}
  $1$ & $(P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)$
  & {\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{hilb27:2}
  $2$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb27:3}
  $3$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb27:2]{$2$}
} \\
\label{hilb27:4}
  $4$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb27:3]{$3$}
} \\
\label{hilb27:5}
  $5$ & $(B\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $(P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb27:4]{$4$}
} \\
\label{hilb27:6}
  $6$ & $((P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \vee \  A)$ in \hyperref[hilb27:5]{$5$}
} \\
\label{hilb27:7}
  $7$ & $\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:1]{$1$}, \hyperref[hilb27:6]{$6$}} \\
\label{hilb27:8}
  $8$ & $P\ \rightarrow \  \neg \neg P$
  & {\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{hilb27:9}
  $9$ & $A\ \rightarrow \  \neg \neg A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb27:8]{$8$}
} \\
\label{hilb27:10}
  $10$ & $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb27:9]{$9$}
} \\
\label{hilb27:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb27:12}
  $12$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb27:11]{$11$}
} \\
\label{hilb27:13}
  $13$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb27:12]{$12$}
} \\
\label{hilb27:14}
  $14$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb27:13]{$13$}
} \\
\label{hilb27:15}
  $15$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb27:14]{$14$}
} \\
\label{hilb27:16}
  $16$ & $(D\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb27:15]{$15$}
} \\
\label{hilb27:17}
  $17$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  Q$ in \hyperref[hilb27:16]{$16$}
} \\
\label{hilb27:18}
  $18$ & $(A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:10]{$10$}, \hyperref[hilb27:17]{$17$}} \\
\label{hilb27:19}
  $19$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb27:20}
  $20$ & $(P\ \vee \  B)\ \rightarrow \  (B\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb27:19]{$19$}
} \\
\label{hilb27:21}
  $21$ & $(C\ \vee \  B)\ \rightarrow \  (B\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb27:20]{$20$}
} \\
\label{hilb27:22}
  $22$ & $(C\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb27:21]{$21$}
} \\
\label{hilb27:23}
  $23$ & $(A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $A$ in \hyperref[hilb27:22]{$22$}
} \\
\label{hilb27:24}
  $24$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb27:25}
  $25$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb27:24]{$24$}
} \\
\label{hilb27:26}
  $26$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb27:25]{$25$}
} \\
\label{hilb27:27}
  $27$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb27:26]{$26$}
} \\
\label{hilb27:28}
  $28$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb27:27]{$27$}
} \\
\label{hilb27:29}
  $29$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb27:28]{$28$}
} \\
\label{hilb27:30}
  $30$ & $((A\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb27:29]{$29$}
} \\
\label{hilb27:31}
  $31$ & $((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (A\ \vee \  \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:23]{$23$}, \hyperref[hilb27:30]{$30$}} \\
\label{hilb27:32}
  $32$ & $(A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:18]{$18$}, \hyperref[hilb27:31]{$31$}} \\
\label{hilb27:33}
  $33$ & $(C\ \vee \  A)\ \rightarrow \  (A\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb27:21]{$21$}
} \\
\label{hilb27:34}
  $34$ & $((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb27:33]{$33$}
} \\
\label{hilb27:35}
  $35$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb27:27]{$27$}
} \\
\label{hilb27:36}
  $36$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb27:35]{$35$}
} \\
\label{hilb27:37}
  $37$ & $((A\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb27:36]{$36$}
} \\
\label{hilb27:38}
  $38$ & $(((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:32]{$32$}, \hyperref[hilb27:37]{$37$}} \\
\label{hilb27:39}
  $39$ & $((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:34]{$34$}, \hyperref[hilb27:38]{$38$}} \\
\label{hilb27:40}
  $40$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb27:2]{$2$}
} \\
\label{hilb27:41}
  $41$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb27:40]{$40$}
} \\
\label{hilb27:42}
  $42$ & $(C\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb27:41]{$41$}
} \\
\label{hilb27:43}
  $43$ & $(((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \vee \  A$ in \hyperref[hilb27:42]{$42$}
} \\
\label{hilb27:44}
  $44$ & $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:39]{$39$}, \hyperref[hilb27:43]{$43$}} \\
\label{hilb27:45}
  $45$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb27:46}
  $46$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb27:47}
  $47$ & $(C\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:41]{$41$}
} \\
\label{hilb27:48}
  $48$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:47]{$47$}
} \\
\label{hilb27:49}
  $49$ & $\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:44]{$44$}, \hyperref[hilb27:48]{$48$}} \\
\label{hilb27:50}
  $50$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:14]{$14$}
} \\
\label{hilb27:51}
  $51$ & $(D\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:50]{$50$}
} \\
\label{hilb27:52}
  $52$ & $(\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:51]{$51$}
} \\
\label{hilb27:53}
  $53$ & $(\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:49]{$49$}, \hyperref[hilb27:52]{$52$}} \\
\label{hilb27:54}
  $54$ & $(C\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:21]{$21$}
} \\
\label{hilb27:55}
  $55$ & $(\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:54]{$54$}
} \\
\label{hilb27:56}
  $56$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:27]{$27$}
} \\
\label{hilb27:57}
  $57$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:56]{$56$}
} \\
\label{hilb27:58}
  $58$ & $((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:57]{$57$}
} \\
\label{hilb27:59}
  $59$ & $((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:55]{$55$}, \hyperref[hilb27:58]{$58$}} \\
\label{hilb27:60}
  $60$ & $(\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:53]{$53$}, \hyperref[hilb27:59]{$59$}} \\
\label{hilb27:61}
  $61$ & $(C\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:21]{$21$}
} \\
\label{hilb27:62}
  $62$ & $(\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:61]{$61$}
} \\
\label{hilb27:63}
  $63$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:27]{$27$}
} \\
\label{hilb27:64}
  $64$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:63]{$63$}
} \\
\label{hilb27:65}
  $65$ & $((\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:64]{$64$}
} \\
\label{hilb27:66}
  $66$ & $((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \vee \  A)))\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:60]{$60$}, \hyperref[hilb27:65]{$65$}} \\
\label{hilb27:67}
  $67$ & $(\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:62]{$62$}, \hyperref[hilb27:66]{$66$}} \\
\label{hilb27:68}
  $68$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb27:45]{$45$}
} \\
\label{hilb27:69}
  $69$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb27:68]{$68$}
} \\
\label{hilb27:70}
  $70$ & $(C\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg C\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:69]{$69$}
} \\
\label{hilb27:71}
  $71$ & $(\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:70]{$70$}
} \\
\label{hilb27:72}
  $72$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:27]{$27$}
} \\
\label{hilb27:73}
  $73$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:72]{$72$}
} \\
\label{hilb27:74}
  $74$ & $((\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:73]{$73$}
} \\
\label{hilb27:75}
  $75$ & $((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:67]{$67$}, \hyperref[hilb27:74]{$74$}} \\
\label{hilb27:76}
  $76$ & $(\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:71]{$71$}, \hyperref[hilb27:75]{$75$}} \\
\label{hilb27:77}
  $77$ & $(\neg P\ \vee \  B)\ \rightarrow \  (P\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb27:46]{$46$}
} \\
\label{hilb27:78}
  $78$ & $(\neg C\ \vee \  B)\ \rightarrow \  (C\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb27:77]{$77$}
} \\
\label{hilb27:79}
  $79$ & $(\neg C\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (C\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:78]{$78$}
} \\
\label{hilb27:80}
  $80$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:79]{$79$}
} \\
\label{hilb27:81}
  $81$ & $(D\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:72]{$72$}
} \\
\label{hilb27:82}
  $82$ & $((\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:81]{$81$}
} \\
\label{hilb27:83}
  $83$ & $((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:80]{$80$}, \hyperref[hilb27:82]{$82$}} \\
\label{hilb27:84}
  $84$ & $(\neg ((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:76]{$76$}, \hyperref[hilb27:83]{$83$}} \\
\label{hilb27:85}
  $85$ & $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:7]{$7$}, \hyperref[hilb27:84]{$84$}} \\
\label{hilb27:86}
  $86$ & $\neg \neg P\ \rightarrow \  P$
  & {\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{hilb27:87}
  $87$ & $\neg \neg B\ \rightarrow \  B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb27:86]{$86$}
} \\
\label{hilb27:88}
  $88$ & $\neg \neg (Q\ \vee \  A)\ \rightarrow \  (Q\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  A$ in \hyperref[hilb27:87]{$87$}
} \\
\label{hilb27:89}
  $89$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb27:14]{$14$}
} \\
\label{hilb27:90}
  $90$ & $(D\ \rightarrow \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q\ \vee \  A$ in \hyperref[hilb27:89]{$89$}
} \\
\label{hilb27:91}
  $91$ & $(\neg \neg (Q\ \vee \  A)\ \rightarrow \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (Q\ \vee \  A)$ in \hyperref[hilb27:90]{$90$}
} \\
\label{hilb27:92}
  $92$ & $(P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:88]{$88$}, \hyperref[hilb27:91]{$91$}} \\
\label{hilb27:93}
  $93$ & $(C\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \vee \  A)$ in \hyperref[hilb27:41]{$41$}
} \\
\label{hilb27:94}
  $94$ & $((P\ \vee \  \neg \neg (Q\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  \neg \neg (Q\ \vee \  A)$ in \hyperref[hilb27:93]{$93$}
} \\
\label{hilb27:95}
  $95$ & $\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:92]{$92$}, \hyperref[hilb27:94]{$94$}} \\
\label{hilb27:96}
  $96$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:14]{$14$}
} \\
\label{hilb27:97}
  $97$ & $(D\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  ((\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb27:96]{$96$}
} \\
\label{hilb27:98}
  $98$ & $(\neg (P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  ((\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:97]{$97$}
} \\
\label{hilb27:99}
  $99$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:95]{$95$}, \hyperref[hilb27:98]{$98$}} \\
\label{hilb27:100}
  $100$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:70]{$70$}
} \\
\label{hilb27:101}
  $101$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:27]{$27$}
} \\
\label{hilb27:102}
  $102$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb27:101]{$101$}
} \\
\label{hilb27:103}
  $103$ & $((\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))$ in \hyperref[hilb27:102]{$102$}
} \\
\label{hilb27:104}
  $104$ & $((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:99]{$99$}, \hyperref[hilb27:103]{$103$}} \\
\label{hilb27:105}
  $105$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:100]{$100$}, \hyperref[hilb27:104]{$104$}} \\
\label{hilb27:106}
  $106$ & $(\neg C\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (C\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb27:78]{$78$}
} \\
\label{hilb27:107}
  $107$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)$ in \hyperref[hilb27:106]{$106$}
} \\
\label{hilb27:108}
  $108$ & $(D\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb27:101]{$101$}
} \\
\label{hilb27:109}
  $109$ & $((\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$ in \hyperref[hilb27:108]{$108$}
} \\
\label{hilb27:110}
  $110$ & $((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:107]{$107$}, \hyperref[hilb27:109]{$109$}} \\
\label{hilb27:111}
  $111$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:105]{$105$}, \hyperref[hilb27:110]{$110$}} \\
\label{hilb27:112}
  $112$ & $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb27:85]{$85$}, \hyperref[hilb27:111]{$111$}} \\
\label{hilb27:113}
  $113$ & $\neg (\neg \neg (P\ \vee \  Q)\ \vee \  B)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb27:112]{$112$}
} \\
\label{hilb27:114}
  $114$ & $\neg (\neg \neg (P\ \vee \  C)\ \vee \  B)\ \rightarrow \  \neg (P\ \vee \  \neg \neg (C\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb27:113]{$113$}
} \\
\label{hilb27:115}
  $115$ & $\neg (\neg \neg (D\ \vee \  C)\ \vee \  B)\ \rightarrow \  \neg (D\ \vee \  \neg \neg (C\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb27:114]{$114$}
} \\
\label{hilb27:116}
  $116$ & $\neg (\neg \neg (D\ \vee \  C)\ \vee \  \neg A)\ \rightarrow \  \neg (D\ \vee \  \neg \neg (C\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg A$ in \hyperref[hilb27:115]{$115$}
} \\
\label{hilb27:117}
  $117$ & $\neg (\neg \neg (D\ \vee \  \neg Q)\ \vee \  \neg A)\ \rightarrow \  \neg (D\ \vee \  \neg \neg (\neg Q\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg Q$ in \hyperref[hilb27:116]{$116$}
} \\
\label{hilb27:118}
  $118$ & $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg A)\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb27:117]{$117$}
} \\
\label{hilb27:119}
  $119$ & $(\neg (\neg P\ \vee \  \neg Q)\ \wedge \  A)\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \vee \  \neg A))$
  & {\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[hilb27:118]{$118$} at occurence $1$
} \\
\label{hilb27:120}
  $120$ & $((P\ \wedge \  Q)\ \wedge \  A)\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \vee \  \neg A))$
  & {\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[hilb27:119]{$119$} at occurence $1$
} \\
\label{hilb27:121}
  $121$ & $((P\ \wedge \  Q)\ \wedge \  A)\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \vee \  \neg A))$
  & {\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[hilb27:120]{$120$} at occurence $1$
} \\
\label{hilb27:122}
  $122$ & $((P\ \wedge \  Q)\ \wedge \  A)\ \rightarrow \  (P\ \wedge \  (Q\ \wedge \  A))$
  & {\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[hilb27:121]{$121$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Form for the conjunction rule:




\begin{thm}[hilb28]
\hypertarget{hilb28}{}
\begin{displaymath}
P\ \rightarrow \  (Q\ \rightarrow \  (P\ \wedge \  Q))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb28:1}
  $1$ & $P\ \vee \  \neg P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb4}{hilb4} } \\
\label{hilb28:2}
  $2$ & $(\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb28:1]{$1$}
} \\
\label{hilb28:3}
  $3$ & $((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb15}{hilb15} } \\
\label{hilb28:4}
  $4$ & $((P\ \vee \  Q)\ \vee \  B)\ \rightarrow \  (P\ \vee \  (Q\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb28:3]{$3$}
} \\
\label{hilb28:5}
  $5$ & $((P\ \vee \  C)\ \vee \  B)\ \rightarrow \  (P\ \vee \  (C\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb28:4]{$4$}
} \\
\label{hilb28:6}
  $6$ & $((D\ \vee \  C)\ \vee \  B)\ \rightarrow \  (D\ \vee \  (C\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb28:5]{$5$}
} \\
\label{hilb28:7}
  $7$ & $((D\ \vee \  C)\ \vee \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (D\ \vee \  (C\ \vee \  \neg (\neg P\ \vee \  \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb28:6]{$6$}
} \\
\label{hilb28:8}
  $8$ & $((D\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (D\ \vee \  (\neg Q\ \vee \  \neg (\neg P\ \vee \  \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg Q$ in \hyperref[hilb28:7]{$7$}
} \\
\label{hilb28:9}
  $9$ & $((\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  \neg (\neg P\ \vee \  \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb28:8]{$8$}
} \\
\label{hilb28:10}
  $10$ & $\neg P\ \vee \  (\neg Q\ \vee \  \neg (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb28:2]{$2$}, \hyperref[hilb28:9]{$9$}} \\
\label{hilb28:11}
  $11$ & $P\ \rightarrow \  (\neg Q\ \vee \  \neg (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb28:10]{$10$} at occurence $1$
} \\
\label{hilb28:12}
  $12$ & $P\ \rightarrow \  (Q\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb28:11]{$11$} at occurence $1$
} \\
\label{hilb28:13}
  $13$ & $P\ \rightarrow \  (Q\ \rightarrow \  (P\ \wedge \  Q))$
  & {\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[hilb28:12]{$12$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Preconditions could be put together in a conjunction (first direction):




\begin{thm}[hilb29]
\hypertarget{hilb29}{}
\begin{displaymath}
(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  A)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb29:1}
  $1$ & $P\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb2}{hilb2} } \\
\label{hilb29:2}
  $2$ & $Q\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb29:1]{$1$}
} \\
\label{hilb29:3}
  $3$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb29:2]{$2$}
} \\
\label{hilb29:4}
  $4$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule5}{use abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb29:3]{$3$} at occurence $4$
} \\
\label{hilb29:5}
  $5$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule5}{use abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb29:4]{$4$} at occurence $4$
} \\
\label{hilb29:6}
  $6$ & $(P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)$
  & {\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{hilb29:7}
  $7$ & $(P\ \vee \  (Q\ \vee \  B))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb29:6]{$6$}
} \\
\label{hilb29:8}
  $8$ & $(P\ \vee \  (C\ \vee \  B))\ \rightarrow \  ((P\ \vee \  C)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb29:7]{$7$}
} \\
\label{hilb29:9}
  $9$ & $(D\ \vee \  (C\ \vee \  B))\ \rightarrow \  ((D\ \vee \  C)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb29:8]{$8$}
} \\
\label{hilb29:10}
  $10$ & $(D\ \vee \  (C\ \vee \  A))\ \rightarrow \  ((D\ \vee \  C)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb29:9]{$9$}
} \\
\label{hilb29:11}
  $11$ & $(D\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  ((D\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg Q$ in \hyperref[hilb29:10]{$10$}
} \\
\label{hilb29:12}
  $12$ & $(\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb29:11]{$11$}
} \\
\label{hilb29:13}
  $13$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb29:14}
  $14$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb29:15}
  $15$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb29:16}
  $16$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb29:15]{$15$}
} \\
\label{hilb29:17}
  $17$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb29:16]{$16$}
} \\
\label{hilb29:18}
  $18$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb29:17]{$17$}
} \\
\label{hilb29:19}
  $19$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb29:18]{$18$}
} \\
\label{hilb29:20}
  $20$ & $(D\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:19]{$19$}
} \\
\label{hilb29:21}
  $21$ & $((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P\ \vee \  (\neg Q\ \vee \  A)$ in \hyperref[hilb29:20]{$20$}
} \\
\label{hilb29:22}
  $22$ & $(\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:12]{$12$}, \hyperref[hilb29:21]{$21$}} \\
\label{hilb29:23}
  $23$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb29:13]{$13$}
} \\
\label{hilb29:24}
  $24$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb29:23]{$23$}
} \\
\label{hilb29:25}
  $25$ & $(C\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg C\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P\ \vee \  (\neg Q\ \vee \  A)$ in \hyperref[hilb29:24]{$24$}
} \\
\label{hilb29:26}
  $26$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb29:25]{$25$}
} \\
\label{hilb29:27}
  $27$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb29:28}
  $28$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb29:27]{$27$}
} \\
\label{hilb29:29}
  $29$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb29:28]{$28$}
} \\
\label{hilb29:30}
  $30$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb29:29]{$29$}
} \\
\label{hilb29:31}
  $31$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A))$ in \hyperref[hilb29:30]{$30$}
} \\
\label{hilb29:32}
  $32$ & $(D\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:31]{$31$}
} \\
\label{hilb29:33}
  $33$ & $((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A))))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A))$ in \hyperref[hilb29:32]{$32$}
} \\
\label{hilb29:34}
  $34$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg P\ \vee \  (\neg Q\ \vee \  A))))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:22]{$22$}, \hyperref[hilb29:33]{$33$}} \\
\label{hilb29:35}
  $35$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:26]{$26$}, \hyperref[hilb29:34]{$34$}} \\
\label{hilb29:36}
  $36$ & $(\neg P\ \vee \  B)\ \rightarrow \  (P\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb29:14]{$14$}
} \\
\label{hilb29:37}
  $37$ & $(\neg C\ \vee \  B)\ \rightarrow \  (C\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb29:36]{$36$}
} \\
\label{hilb29:38}
  $38$ & $(\neg C\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (C\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:37]{$37$}
} \\
\label{hilb29:39}
  $39$ & $(\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb29:38]{$38$}
} \\
\label{hilb29:40}
  $40$ & $(D\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:31]{$31$}
} \\
\label{hilb29:41}
  $41$ & $((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:40]{$40$}
} \\
\label{hilb29:42}
  $42$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:39]{$39$}, \hyperref[hilb29:41]{$41$}} \\
\label{hilb29:43}
  $43$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:35]{$35$}, \hyperref[hilb29:42]{$42$}} \\
\label{hilb29:44}
  $44$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:5]{$5$}, \hyperref[hilb29:43]{$43$}} \\
\label{hilb29:45}
  $45$ & $P\ \rightarrow \  \neg \neg P$
  & {\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{hilb29:46}
  $46$ & $A\ \rightarrow \  \neg \neg A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb29:45]{$45$}
} \\
\label{hilb29:47}
  $47$ & $(\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg \neg (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb29:46]{$46$}
} \\
\label{hilb29:48}
  $48$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb29:18]{$18$}
} \\
\label{hilb29:49}
  $49$ & $(D\ \rightarrow \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb29:48]{$48$}
} \\
\label{hilb29:50}
  $50$ & $((\neg P\ \vee \  \neg Q)\ \rightarrow \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb29:49]{$49$}
} \\
\label{hilb29:51}
  $51$ & $(A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:47]{$47$}, \hyperref[hilb29:50]{$50$}} \\
\label{hilb29:52}
  $52$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb29:53}
  $53$ & $(P\ \vee \  B)\ \rightarrow \  (B\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb29:52]{$52$}
} \\
\label{hilb29:54}
  $54$ & $(C\ \vee \  B)\ \rightarrow \  (B\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb29:53]{$53$}
} \\
\label{hilb29:55}
  $55$ & $(C\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb29:54]{$54$}
} \\
\label{hilb29:56}
  $56$ & $(A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $A$ in \hyperref[hilb29:55]{$55$}
} \\
\label{hilb29:57}
  $57$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A\ \vee \  (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb29:30]{$30$}
} \\
\label{hilb29:58}
  $58$ & $(D\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:57]{$57$}
} \\
\label{hilb29:59}
  $59$ & $((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)))\ \rightarrow \  ((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb29:58]{$58$}
} \\
\label{hilb29:60}
  $60$ & $((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)))\ \rightarrow \  ((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:56]{$56$}, \hyperref[hilb29:59]{$59$}} \\
\label{hilb29:61}
  $61$ & $(A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:51]{$51$}, \hyperref[hilb29:60]{$60$}} \\
\label{hilb29:62}
  $62$ & $(C\ \vee \  A)\ \rightarrow \  (A\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb29:54]{$54$}
} \\
\label{hilb29:63}
  $63$ & $((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb29:62]{$62$}
} \\
\label{hilb29:64}
  $64$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:30]{$30$}
} \\
\label{hilb29:65}
  $65$ & $(D\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:64]{$64$}
} \\
\label{hilb29:66}
  $66$ & $((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb29:65]{$65$}
} \\
\label{hilb29:67}
  $67$ & $(((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:61]{$61$}, \hyperref[hilb29:66]{$66$}} \\
\label{hilb29:68}
  $68$ & $((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:63]{$63$}, \hyperref[hilb29:67]{$67$}} \\
\label{hilb29:69}
  $69$ & $(D\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:19]{$19$}
} \\
\label{hilb29:70}
  $70$ & $(((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:69]{$69$}
} \\
\label{hilb29:71}
  $71$ & $(\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:68]{$68$}, \hyperref[hilb29:70]{$70$}} \\
\label{hilb29:72}
  $72$ & $(C\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg C\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:24]{$24$}
} \\
\label{hilb29:73}
  $73$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb29:72]{$72$}
} \\
\label{hilb29:74}
  $74$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:30]{$30$}
} \\
\label{hilb29:75}
  $75$ & $(D\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:74]{$74$}
} \\
\label{hilb29:76}
  $76$ & $((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:75]{$75$}
} \\
\label{hilb29:77}
  $77$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:71]{$71$}, \hyperref[hilb29:76]{$76$}} \\
\label{hilb29:78}
  $78$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:73]{$73$}, \hyperref[hilb29:77]{$77$}} \\
\label{hilb29:79}
  $79$ & $(\neg C\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (C\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb29:37]{$37$}
} \\
\label{hilb29:80}
  $80$ & $(\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb29:79]{$79$}
} \\
\label{hilb29:81}
  $81$ & $(D\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:74]{$74$}
} \\
\label{hilb29:82}
  $82$ & $((\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb29:81]{$81$}
} \\
\label{hilb29:83}
  $83$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:80]{$80$}, \hyperref[hilb29:82]{$82$}} \\
\label{hilb29:84}
  $84$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:78]{$78$}, \hyperref[hilb29:83]{$83$}} \\
\label{hilb29:85}
  $85$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb29:44]{$44$}, \hyperref[hilb29:84]{$84$}} \\
\label{hilb29:86}
  $86$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb29:85]{$85$} at occurence $1$
} \\
\label{hilb29:87}
  $87$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  A)$
  & {\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[hilb29:86]{$86$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Preconditions could be put together in a conjunction (second direction):




\begin{thm}[hilb30]
\hypertarget{hilb30}{}
\begin{displaymath}
((P\ \wedge \  Q)\ \rightarrow \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb30:1}
  $1$ & $P\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb2}{hilb2} } \\
\label{hilb30:2}
  $2$ & $Q\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb30:1]{$1$}
} \\
\label{hilb30:3}
  $3$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb30:2]{$2$}
} \\
\label{hilb30:4}
  $4$ & $(\neg P\ \vee \  (Q\ \rightarrow \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule5}{use abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb30:3]{$3$} at occurence $2$
} \\
\label{hilb30:5}
  $5$ & $(\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule5}{use abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb30:4]{$4$} at occurence $2$
} \\
\label{hilb30:6}
  $6$ & $((P\ \vee \  Q)\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb15}{hilb15} } \\
\label{hilb30:7}
  $7$ & $((P\ \vee \  Q)\ \vee \  B)\ \rightarrow \  (P\ \vee \  (Q\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb30:6]{$6$}
} \\
\label{hilb30:8}
  $8$ & $((P\ \vee \  C)\ \vee \  B)\ \rightarrow \  (P\ \vee \  (C\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb30:7]{$7$}
} \\
\label{hilb30:9}
  $9$ & $((D\ \vee \  C)\ \vee \  B)\ \rightarrow \  (D\ \vee \  (C\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb30:8]{$8$}
} \\
\label{hilb30:10}
  $10$ & $((D\ \vee \  C)\ \vee \  A)\ \rightarrow \  (D\ \vee \  (C\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb30:9]{$9$}
} \\
\label{hilb30:11}
  $11$ & $((D\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (D\ \vee \  (\neg Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg Q$ in \hyperref[hilb30:10]{$10$}
} \\
\label{hilb30:12}
  $12$ & $((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb30:11]{$11$}
} \\
\label{hilb30:13}
  $13$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb30:14}
  $14$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb30:15}
  $15$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb30:16}
  $16$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb30:15]{$15$}
} \\
\label{hilb30:17}
  $17$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb30:16]{$16$}
} \\
\label{hilb30:18}
  $18$ & $(C\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P\ \vee \  (\neg Q\ \vee \  A)$ in \hyperref[hilb30:17]{$17$}
} \\
\label{hilb30:19}
  $19$ & $(((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:18]{$18$}
} \\
\label{hilb30:20}
  $20$ & $\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:12]{$12$}, \hyperref[hilb30:19]{$19$}} \\
\label{hilb30:21}
  $21$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb30:22}
  $22$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb30:21]{$21$}
} \\
\label{hilb30:23}
  $23$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb30:22]{$22$}
} \\
\label{hilb30:24}
  $24$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb30:23]{$23$}
} \\
\label{hilb30:25}
  $25$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  D)\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb30:24]{$24$}
} \\
\label{hilb30:26}
  $26$ & $(D\ \rightarrow \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  D)\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:25]{$25$}
} \\
\label{hilb30:27}
  $27$ & $(\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))$ in \hyperref[hilb30:26]{$26$}
} \\
\label{hilb30:28}
  $28$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:20]{$20$}, \hyperref[hilb30:27]{$27$}} \\
\label{hilb30:29}
  $29$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb30:30}
  $30$ & $(P\ \vee \  B)\ \rightarrow \  (B\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb30:29]{$29$}
} \\
\label{hilb30:31}
  $31$ & $(C\ \vee \  B)\ \rightarrow \  (B\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb30:30]{$30$}
} \\
\label{hilb30:32}
  $32$ & $(C\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:31]{$31$}
} \\
\label{hilb30:33}
  $33$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb30:32]{$32$}
} \\
\label{hilb30:34}
  $34$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb30:35}
  $35$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb30:34]{$34$}
} \\
\label{hilb30:36}
  $36$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb30:35]{$35$}
} \\
\label{hilb30:37}
  $37$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb30:36]{$36$}
} \\
\label{hilb30:38}
  $38$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A))$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:39}
  $39$ & $(D\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:38]{$38$}
} \\
\label{hilb30:40}
  $40$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:39]{$39$}
} \\
\label{hilb30:41}
  $41$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:33]{$33$}, \hyperref[hilb30:40]{$40$}} \\
\label{hilb30:42}
  $42$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:28]{$28$}, \hyperref[hilb30:41]{$41$}} \\
\label{hilb30:43}
  $43$ & $(C\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb30:31]{$31$}
} \\
\label{hilb30:44}
  $44$ & $(\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))$ in \hyperref[hilb30:43]{$43$}
} \\
\label{hilb30:45}
  $45$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:46}
  $46$ & $(D\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:45]{$45$}
} \\
\label{hilb30:47}
  $47$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A))$ in \hyperref[hilb30:46]{$46$}
} \\
\label{hilb30:48}
  $48$ & $((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg P\ \vee \  (\neg Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:42]{$42$}, \hyperref[hilb30:47]{$47$}} \\
\label{hilb30:49}
  $49$ & $(\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:44]{$44$}, \hyperref[hilb30:48]{$48$}} \\
\label{hilb30:50}
  $50$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb30:13]{$13$}
} \\
\label{hilb30:51}
  $51$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb30:50]{$50$}
} \\
\label{hilb30:52}
  $52$ & $(C\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg C\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb30:51]{$51$}
} \\
\label{hilb30:53}
  $53$ & $((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  (\neg Q\ \vee \  A)$ in \hyperref[hilb30:52]{$52$}
} \\
\label{hilb30:54}
  $54$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  (((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:55}
  $55$ & $(D\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  (((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:54]{$54$}
} \\
\label{hilb30:56}
  $56$ & $((\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:55]{$55$}
} \\
\label{hilb30:57}
  $57$ & $(((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg P\ \vee \  (\neg Q\ \vee \  A))\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:49]{$49$}, \hyperref[hilb30:56]{$56$}} \\
\label{hilb30:58}
  $58$ & $((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:53]{$53$}, \hyperref[hilb30:57]{$57$}} \\
\label{hilb30:59}
  $59$ & $(\neg P\ \vee \  B)\ \rightarrow \  (P\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb30:14]{$14$}
} \\
\label{hilb30:60}
  $60$ & $(\neg C\ \vee \  B)\ \rightarrow \  (C\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb30:59]{$59$}
} \\
\label{hilb30:61}
  $61$ & $(\neg C\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (C\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb30:60]{$60$}
} \\
\label{hilb30:62}
  $62$ & $(\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:61]{$61$}
} \\
\label{hilb30:63}
  $63$ & $(D\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  (((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:54]{$54$}
} \\
\label{hilb30:64}
  $64$ & $((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:63]{$63$}
} \\
\label{hilb30:65}
  $65$ & $(((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:62]{$62$}, \hyperref[hilb30:64]{$64$}} \\
\label{hilb30:66}
  $66$ & $((\neg P\ \vee \  (\neg Q\ \vee \  A))\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:58]{$58$}, \hyperref[hilb30:65]{$65$}} \\
\label{hilb30:67}
  $67$ & $((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:5]{$5$}, \hyperref[hilb30:66]{$66$}} \\
\label{hilb30:68}
  $68$ & $\neg \neg P\ \rightarrow \  P$
  & {\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{hilb30:69}
  $69$ & $\neg \neg A\ \rightarrow \  A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb30:68]{$68$}
} \\
\label{hilb30:70}
  $70$ & $\neg \neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb30:69]{$69$}
} \\
\label{hilb30:71}
  $71$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb30:24]{$24$}
} \\
\label{hilb30:72}
  $72$ & $(D\ \rightarrow \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((A\ \vee \  D)\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb30:71]{$71$}
} \\
\label{hilb30:73}
  $73$ & $(\neg \neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb30:72]{$72$}
} \\
\label{hilb30:74}
  $74$ & $(A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:70]{$70$}, \hyperref[hilb30:73]{$73$}} \\
\label{hilb30:75}
  $75$ & $(C\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P\ \vee \  \neg Q$ in \hyperref[hilb30:31]{$31$}
} \\
\label{hilb30:76}
  $76$ & $(A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $A$ in \hyperref[hilb30:75]{$75$}
} \\
\label{hilb30:77}
  $77$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:78}
  $78$ & $(D\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  D)\ \rightarrow \  ((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:77]{$77$}
} \\
\label{hilb30:79}
  $79$ & $((A\ \vee \  (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q)))\ \rightarrow \  ((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb30:78]{$78$}
} \\
\label{hilb30:80}
  $80$ & $((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  (A\ \vee \  (\neg P\ \vee \  \neg Q)))\ \rightarrow \  ((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:76]{$76$}, \hyperref[hilb30:79]{$79$}} \\
\label{hilb30:81}
  $81$ & $(A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:74]{$74$}, \hyperref[hilb30:80]{$80$}} \\
\label{hilb30:82}
  $82$ & $(C\ \vee \  A)\ \rightarrow \  (A\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb30:31]{$31$}
} \\
\label{hilb30:83}
  $83$ & $(\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb30:82]{$82$}
} \\
\label{hilb30:84}
  $84$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:85}
  $85$ & $(D\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:84]{$84$}
} \\
\label{hilb30:86}
  $86$ & $((A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q))\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb30:85]{$85$}
} \\
\label{hilb30:87}
  $87$ & $((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (A\ \vee \  \neg \neg (\neg P\ \vee \  \neg Q)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:81]{$81$}, \hyperref[hilb30:86]{$86$}} \\
\label{hilb30:88}
  $88$ & $(\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:83]{$83$}, \hyperref[hilb30:87]{$87$}} \\
\label{hilb30:89}
  $89$ & $(C\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:17]{$17$}
} \\
\label{hilb30:90}
  $90$ & $((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:89]{$89$}
} \\
\label{hilb30:91}
  $91$ & $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:88]{$88$}, \hyperref[hilb30:90]{$90$}} \\
\label{hilb30:92}
  $92$ & $(D\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  D)\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:25]{$25$}
} \\
\label{hilb30:93}
  $93$ & $(\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:92]{$92$}
} \\
\label{hilb30:94}
  $94$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:91]{$91$}, \hyperref[hilb30:93]{$93$}} \\
\label{hilb30:95}
  $95$ & $(C\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:31]{$31$}
} \\
\label{hilb30:96}
  $96$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  (Q\ \rightarrow \  A)$ in \hyperref[hilb30:95]{$95$}
} \\
\label{hilb30:97}
  $97$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:98}
  $98$ & $(D\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:97]{$97$}
} \\
\label{hilb30:99}
  $99$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:98]{$98$}
} \\
\label{hilb30:100}
  $100$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  (((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:96]{$96$}, \hyperref[hilb30:99]{$99$}} \\
\label{hilb30:101}
  $101$ & $((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:94]{$94$}, \hyperref[hilb30:100]{$100$}} \\
\label{hilb30:102}
  $102$ & $(\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:43]{$43$}
} \\
\label{hilb30:103}
  $103$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:104}
  $104$ & $(D\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:103]{$103$}
} \\
\label{hilb30:105}
  $105$ & $(((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)$ in \hyperref[hilb30:104]{$104$}
} \\
\label{hilb30:106}
  $106$ & $((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((P\ \rightarrow \  (Q\ \rightarrow \  A))\ \vee \  \neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)))\ \rightarrow \  ((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:101]{$101$}, \hyperref[hilb30:105]{$105$}} \\
\label{hilb30:107}
  $107$ & $(\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:102]{$102$}, \hyperref[hilb30:106]{$106$}} \\
\label{hilb30:108}
  $108$ & $(((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:52]{$52$}
} \\
\label{hilb30:109}
  $109$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:37]{$37$}
} \\
\label{hilb30:110}
  $110$ & $(D\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:109]{$109$}
} \\
\label{hilb30:111}
  $111$ & $((\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:110]{$110$}
} \\
\label{hilb30:112}
  $112$ & $((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg ((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:107]{$107$}, \hyperref[hilb30:111]{$111$}} \\
\label{hilb30:113}
  $113$ & $(((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:108]{$108$}, \hyperref[hilb30:112]{$112$}} \\
\label{hilb30:114}
  $114$ & $(\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A$ in \hyperref[hilb30:61]{$61$}
} \\
\label{hilb30:115}
  $115$ & $(D\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  D)\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:109]{$109$}
} \\
\label{hilb30:116}
  $116$ & $((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  (((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$ in \hyperref[hilb30:115]{$115$}
} \\
\label{hilb30:117}
  $117$ & $((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \vee \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))\ \rightarrow \  ((((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:114]{$114$}, \hyperref[hilb30:116]{$116$}} \\
\label{hilb30:118}
  $118$ & $(((\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))\ \rightarrow \  ((\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:113]{$113$}, \hyperref[hilb30:117]{$117$}} \\
\label{hilb30:119}
  $119$ & $(\neg \neg (\neg P\ \vee \  \neg Q)\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb30:67]{$67$}, \hyperref[hilb30:118]{$118$}} \\
\label{hilb30:120}
  $120$ & $(\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[hilb30:119]{$119$} at occurence $1$
} \\
\label{hilb30:121}
  $121$ & $((P\ \wedge \  Q)\ \rightarrow \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\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[hilb30:120]{$120$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Absorbtion of a conjunction (first direction):




\begin{thm}[hilb31]
\hypertarget{hilb31}{}
\begin{displaymath}
(P\ \wedge \  P)\ \rightarrow \  P\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb31:1}
  $1$ & $(P\ \wedge \  Q)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb24}{hilb24} } \\
\label{hilb31:2}
  $2$ & $(P\ \wedge \  P)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $P$ in \hyperref[hilb31:1]{$1$}
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Absorbtion of a conjunction (second direction):




\begin{thm}[hilb32]
\hypertarget{hilb32}{}
\begin{displaymath}
P\ \rightarrow \  (P\ \wedge \  P)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb32:1}
  $1$ & $(P\ \vee \  P)\ \rightarrow \  P$
  & {\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{hilb32:2}
  $2$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb32:3}
  $3$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb32:2]{$2$}
} \\
\label{hilb32:4}
  $4$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb32:3]{$3$}
} \\
\label{hilb32:5}
  $5$ & $(B\ \rightarrow \  P)\ \rightarrow \  (\neg P\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb32:4]{$4$}
} \\
\label{hilb32:6}
  $6$ & $((P\ \vee \  P)\ \rightarrow \  P)\ \rightarrow \  (\neg P\ \rightarrow \  \neg (P\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  P$ in \hyperref[hilb32:5]{$5$}
} \\
\label{hilb32:7}
  $7$ & $\neg P\ \rightarrow \  \neg (P\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:1]{$1$}, \hyperref[hilb32:6]{$6$}} \\
\label{hilb32:8}
  $8$ & $\neg Q\ \rightarrow \  \neg (Q\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb32:7]{$7$}
} \\
\label{hilb32:9}
  $9$ & $\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $\neg P$ in \hyperref[hilb32:8]{$8$}
} \\
\label{hilb32:10}
  $10$ & $P\ \rightarrow \  \neg \neg P$
  & {\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{hilb32:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb32:12}
  $12$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb32:13}
  $13$ & $(B\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg \neg \neg P\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P$ in \hyperref[hilb32:4]{$4$}
} \\
\label{hilb32:14}
  $14$ & $(P\ \rightarrow \  \neg \neg P)\ \rightarrow \  (\neg \neg \neg P\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb32:13]{$13$}
} \\
\label{hilb32:15}
  $15$ & $\neg \neg \neg P\ \rightarrow \  \neg P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:10]{$10$}, \hyperref[hilb32:14]{$14$}} \\
\label{hilb32:16}
  $16$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb32:17}
  $17$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb32:16]{$16$}
} \\
\label{hilb32:18}
  $18$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb32:17]{$17$}
} \\
\label{hilb32:19}
  $19$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb32:18]{$18$}
} \\
\label{hilb32:20}
  $20$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (\neg P\ \vee \  \neg P)\ \vee \  D)\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:19]{$19$}
} \\
\label{hilb32:21}
  $21$ & $(D\ \rightarrow \  \neg P)\ \rightarrow \  ((\neg (\neg P\ \vee \  \neg P)\ \vee \  D)\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P$ in \hyperref[hilb32:20]{$20$}
} \\
\label{hilb32:22}
  $22$ & $(\neg \neg \neg P\ \rightarrow \  \neg P)\ \rightarrow \  ((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg P$ in \hyperref[hilb32:21]{$21$}
} \\
\label{hilb32:23}
  $23$ & $(\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:15]{$15$}, \hyperref[hilb32:22]{$22$}} \\
\label{hilb32:24}
  $24$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb32:25}
  $25$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb32:24]{$24$}
} \\
\label{hilb32:26}
  $26$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb32:25]{$25$}
} \\
\label{hilb32:27}
  $27$ & $(B\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg P$ in \hyperref[hilb32:26]{$26$}
} \\
\label{hilb32:28}
  $28$ & $(\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:27]{$27$}
} \\
\label{hilb32:29}
  $29$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb32:30}
  $30$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb32:29]{$29$}
} \\
\label{hilb32:31}
  $31$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb32:30]{$30$}
} \\
\label{hilb32:32}
  $32$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb32:31]{$31$}
} \\
\label{hilb32:33}
  $33$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P$ in \hyperref[hilb32:32]{$32$}
} \\
\label{hilb32:34}
  $34$ & $(D\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:33]{$33$}
} \\
\label{hilb32:35}
  $35$ & $((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P))\ \rightarrow \  ((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P$ in \hyperref[hilb32:34]{$34$}
} \\
\label{hilb32:36}
  $36$ & $((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg P))\ \rightarrow \  ((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:28]{$28$}, \hyperref[hilb32:35]{$35$}} \\
\label{hilb32:37}
  $37$ & $(\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:23]{$23$}, \hyperref[hilb32:36]{$36$}} \\
\label{hilb32:38}
  $38$ & $(B\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:26]{$26$}
} \\
\label{hilb32:39}
  $39$ & $(\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg P$ in \hyperref[hilb32:38]{$38$}
} \\
\label{hilb32:40}
  $40$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:32]{$32$}
} \\
\label{hilb32:41}
  $41$ & $(D\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:40]{$40$}
} \\
\label{hilb32:42}
  $42$ & $((\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P)\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P))\ \rightarrow \  ((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P$ in \hyperref[hilb32:41]{$41$}
} \\
\label{hilb32:43}
  $43$ & $((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg P)\ \vee \  \neg \neg \neg P))\ \rightarrow \  ((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:37]{$37$}, \hyperref[hilb32:42]{$42$}} \\
\label{hilb32:44}
  $44$ & $(\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:39]{$39$}, \hyperref[hilb32:43]{$43$}} \\
\label{hilb32:45}
  $45$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb32:11]{$11$}
} \\
\label{hilb32:46}
  $46$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb32:45]{$45$}
} \\
\label{hilb32:47}
  $47$ & $(B\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg B\ \vee \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:46]{$46$}
} \\
\label{hilb32:48}
  $48$ & $(\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P$ in \hyperref[hilb32:47]{$47$}
} \\
\label{hilb32:49}
  $49$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:32]{$32$}
} \\
\label{hilb32:50}
  $50$ & $(D\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:49]{$49$}
} \\
\label{hilb32:51}
  $51$ & $((\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  ((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:50]{$50$}
} \\
\label{hilb32:52}
  $52$ & $((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg \neg \neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  ((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:44]{$44$}, \hyperref[hilb32:51]{$51$}} \\
\label{hilb32:53}
  $53$ & $(\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:48]{$48$}, \hyperref[hilb32:52]{$52$}} \\
\label{hilb32:54}
  $54$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb32:12]{$12$}
} \\
\label{hilb32:55}
  $55$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb32:54]{$54$}
} \\
\label{hilb32:56}
  $56$ & $(\neg B\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (B\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:55]{$55$}
} \\
\label{hilb32:57}
  $57$ & $(\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb32:56]{$56$}
} \\
\label{hilb32:58}
  $58$ & $(D\ \rightarrow \  (P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:49]{$49$}
} \\
\label{hilb32:59}
  $59$ & $((\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  (((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  ((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)$ in \hyperref[hilb32:58]{$58$}
} \\
\label{hilb32:60}
  $60$ & $((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (\neg P\ \vee \  \neg (\neg P\ \vee \  \neg P)))\ \rightarrow \  ((\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:57]{$57$}, \hyperref[hilb32:59]{$59$}} \\
\label{hilb32:61}
  $61$ & $(\neg \neg P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))\ \rightarrow \  (P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:53]{$53$}, \hyperref[hilb32:60]{$60$}} \\
\label{hilb32:62}
  $62$ & $P\ \rightarrow \  \neg (\neg P\ \vee \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb32:9]{$9$}, \hyperref[hilb32:61]{$61$}} \\
\label{hilb32:63}
  $63$ & $P\ \rightarrow \  (P\ \wedge \  P)$
  & {\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[hilb32:62]{$62$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Absorbtion of identical preconditions (first direction):




\begin{thm}[hilb33]
\hypertarget{hilb33}{}
\begin{displaymath}
(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb33:1}
  $1$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb29}{hilb29} } \\
\label{hilb33:2}
  $2$ & $(P\ \rightarrow \  (Q\ \rightarrow \  B))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb33:1]{$1$}
} \\
\label{hilb33:3}
  $3$ & $(P\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  ((P\ \wedge \  C)\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb33:2]{$2$}
} \\
\label{hilb33:4}
  $4$ & $(D\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  ((D\ \wedge \  C)\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb33:3]{$3$}
} \\
\label{hilb33:5}
  $5$ & $(D\ \rightarrow \  (C\ \rightarrow \  Q))\ \rightarrow \  ((D\ \wedge \  C)\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb33:4]{$4$}
} \\
\label{hilb33:6}
  $6$ & $(D\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((D\ \wedge \  P)\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb33:5]{$5$}
} \\
\label{hilb33:7}
  $7$ & $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P$ in \hyperref[hilb33:6]{$6$}
} \\
\label{hilb33:8}
  $8$ & $P\ \rightarrow \  (P\ \wedge \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb32}{hilb32} } \\
\label{hilb33:9}
  $9$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb33:10}
  $10$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb33:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb33:12}
  $12$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb33:11]{$11$}
} \\
\label{hilb33:13}
  $13$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb33:12]{$12$}
} \\
\label{hilb33:14}
  $14$ & $(B\ \rightarrow \  (P\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  P)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \wedge \  P$ in \hyperref[hilb33:13]{$13$}
} \\
\label{hilb33:15}
  $15$ & $(P\ \rightarrow \  (P\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  P)\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb33:14]{$14$}
} \\
\label{hilb33:16}
  $16$ & $\neg (P\ \wedge \  P)\ \rightarrow \  \neg P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:8]{$8$}, \hyperref[hilb33:15]{$15$}} \\
\label{hilb33:17}
  $17$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb33:18}
  $18$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb33:17]{$17$}
} \\
\label{hilb33:19}
  $19$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb33:18]{$18$}
} \\
\label{hilb33:20}
  $20$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb33:19]{$19$}
} \\
\label{hilb33:21}
  $21$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb33:20]{$20$}
} \\
\label{hilb33:22}
  $22$ & $(D\ \rightarrow \  \neg P)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P$ in \hyperref[hilb33:21]{$21$}
} \\
\label{hilb33:23}
  $23$ & $(\neg (P\ \wedge \  P)\ \rightarrow \  \neg P)\ \rightarrow \  ((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  P)$ in \hyperref[hilb33:22]{$22$}
} \\
\label{hilb33:24}
  $24$ & $(Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:16]{$16$}, \hyperref[hilb33:23]{$23$}} \\
\label{hilb33:25}
  $25$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb33:26}
  $26$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb33:25]{$25$}
} \\
\label{hilb33:27}
  $27$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb33:26]{$26$}
} \\
\label{hilb33:28}
  $28$ & $(B\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg P$ in \hyperref[hilb33:27]{$27$}
} \\
\label{hilb33:29}
  $29$ & $(Q\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb33:28]{$28$}
} \\
\label{hilb33:30}
  $30$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb33:31}
  $31$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb33:30]{$30$}
} \\
\label{hilb33:32}
  $32$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb33:31]{$31$}
} \\
\label{hilb33:33}
  $33$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb33:32]{$32$}
} \\
\label{hilb33:34}
  $34$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  \neg (P\ \wedge \  P)$ in \hyperref[hilb33:33]{$33$}
} \\
\label{hilb33:35}
  $35$ & $(D\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  Q$ in \hyperref[hilb33:34]{$34$}
} \\
\label{hilb33:36}
  $36$ & $((Q\ \vee \  \neg P)\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg P))\ \rightarrow \  ((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg P$ in \hyperref[hilb33:35]{$35$}
} \\
\label{hilb33:37}
  $37$ & $((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (Q\ \vee \  \neg P))\ \rightarrow \  ((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:29]{$29$}, \hyperref[hilb33:36]{$36$}} \\
\label{hilb33:38}
  $38$ & $(Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:24]{$24$}, \hyperref[hilb33:37]{$37$}} \\
\label{hilb33:39}
  $39$ & $(B\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb33:27]{$27$}
} \\
\label{hilb33:40}
  $40$ & $(\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \wedge \  P)$ in \hyperref[hilb33:39]{$39$}
} \\
\label{hilb33:41}
  $41$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb33:33]{$33$}
} \\
\label{hilb33:42}
  $42$ & $(D\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  (((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  Q$ in \hyperref[hilb33:41]{$41$}
} \\
\label{hilb33:43}
  $43$ & $((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  (((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P)))\ \rightarrow \  ((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg (P\ \wedge \  P)$ in \hyperref[hilb33:42]{$42$}
} \\
\label{hilb33:44}
  $44$ & $((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P)))\ \rightarrow \  ((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:38]{$38$}, \hyperref[hilb33:43]{$43$}} \\
\label{hilb33:45}
  $45$ & $(\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:40]{$40$}, \hyperref[hilb33:44]{$44$}} \\
\label{hilb33:46}
  $46$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb33:9]{$9$}
} \\
\label{hilb33:47}
  $47$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb33:46]{$46$}
} \\
\label{hilb33:48}
  $48$ & $(B\ \rightarrow \  Q)\ \rightarrow \  (\neg B\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb33:47]{$47$}
} \\
\label{hilb33:49}
  $49$ & $((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  P$ in \hyperref[hilb33:48]{$48$}
} \\
\label{hilb33:50}
  $50$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \wedge \  P)\ \rightarrow \  Q$ in \hyperref[hilb33:33]{$33$}
} \\
\label{hilb33:51}
  $51$ & $(D\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  Q$ in \hyperref[hilb33:50]{$50$}
} \\
\label{hilb33:52}
  $52$ & $((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb33:51]{$51$}
} \\
\label{hilb33:53}
  $53$ & $(((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:45]{$45$}, \hyperref[hilb33:52]{$52$}} \\
\label{hilb33:54}
  $54$ & $((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:49]{$49$}, \hyperref[hilb33:53]{$53$}} \\
\label{hilb33:55}
  $55$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb33:10]{$10$}
} \\
\label{hilb33:56}
  $56$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb33:55]{$55$}
} \\
\label{hilb33:57}
  $57$ & $(D\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  Q$ in \hyperref[hilb33:50]{$50$}
} \\
\label{hilb33:58}
  $58$ & $((\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  (((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P\ \vee \  Q$ in \hyperref[hilb33:57]{$57$}
} \\
\label{hilb33:59}
  $59$ & $(((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  (((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:10]{$10$}, \hyperref[hilb33:58]{$58$}} \\
\label{hilb33:60}
  $60$ & $((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:54]{$54$}, \hyperref[hilb33:59]{$59$}} \\
\label{hilb33:61}
  $61$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  D)\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb33:20]{$20$}
} \\
\label{hilb33:62}
  $62$ & $(D\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  D)\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \rightarrow \  Q$ in \hyperref[hilb33:61]{$61$}
} \\
\label{hilb33:63}
  $63$ & $(((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \wedge \  P)\ \rightarrow \  Q$ in \hyperref[hilb33:62]{$62$}
} \\
\label{hilb33:64}
  $64$ & $(\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:60]{$60$}, \hyperref[hilb33:63]{$63$}} \\
\label{hilb33:65}
  $65$ & $(B\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg B\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $(P\ \wedge \  P)\ \rightarrow \  Q$ in \hyperref[hilb33:47]{$47$}
} \\
\label{hilb33:66}
  $66$ & $((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb33:65]{$65$}
} \\
\label{hilb33:67}
  $67$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q)$ in \hyperref[hilb33:33]{$33$}
} \\
\label{hilb33:68}
  $68$ & $(D\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)$ in \hyperref[hilb33:67]{$67$}
} \\
\label{hilb33:69}
  $69$ & $((\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q)))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q)$ in \hyperref[hilb33:68]{$68$}
} \\
\label{hilb33:70}
  $70$ & $(((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  ((P\ \wedge \  P)\ \rightarrow \  Q)))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:64]{$64$}, \hyperref[hilb33:69]{$69$}} \\
\label{hilb33:71}
  $71$ & $((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:66]{$66$}, \hyperref[hilb33:70]{$70$}} \\
\label{hilb33:72}
  $72$ & $(\neg B\ \vee \  (P\ \rightarrow \  Q))\ \rightarrow \  (B\ \rightarrow \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \rightarrow \  Q$ in \hyperref[hilb33:56]{$56$}
} \\
\label{hilb33:73}
  $73$ & $(\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb33:72]{$72$}
} \\
\label{hilb33:74}
  $74$ & $(D\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb33:67]{$67$}
} \\
\label{hilb33:75}
  $75$ & $((\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)$ in \hyperref[hilb33:74]{$74$}
} \\
\label{hilb33:76}
  $76$ & $(((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  (P\ \rightarrow \  Q)))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:73]{$73$}, \hyperref[hilb33:75]{$75$}} \\
\label{hilb33:77}
  $77$ & $((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:71]{$71$}, \hyperref[hilb33:76]{$76$}} \\
\label{hilb33:78}
  $78$ & $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb33:7]{$7$}, \hyperref[hilb33:77]{$77$}} \\
 & & \qedhere
\end{longtable}
\end{proof}



Absorbtion of identical preconditions (second direction):




\begin{thm}[hilb34]
\hypertarget{hilb34}{}
\begin{displaymath}
(P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb34:1}
  $1$ & $((P\ \wedge \  Q)\ \rightarrow \  A)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb30}{hilb30} } \\
\label{hilb34:2}
  $2$ & $((P\ \wedge \  Q)\ \rightarrow \  B)\ \rightarrow \  (P\ \rightarrow \  (Q\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb34:1]{$1$}
} \\
\label{hilb34:3}
  $3$ & $((P\ \wedge \  C)\ \rightarrow \  B)\ \rightarrow \  (P\ \rightarrow \  (C\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb34:2]{$2$}
} \\
\label{hilb34:4}
  $4$ & $((D\ \wedge \  C)\ \rightarrow \  B)\ \rightarrow \  (D\ \rightarrow \  (C\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb34:3]{$3$}
} \\
\label{hilb34:5}
  $5$ & $((D\ \wedge \  C)\ \rightarrow \  Q)\ \rightarrow \  (D\ \rightarrow \  (C\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb34:4]{$4$}
} \\
\label{hilb34:6}
  $6$ & $((D\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (D\ \rightarrow \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb34:5]{$5$}
} \\
\label{hilb34:7}
  $7$ & $((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P$ in \hyperref[hilb34:6]{$6$}
} \\
\label{hilb34:8}
  $8$ & $(P\ \wedge \  P)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb31}{hilb31} } \\
\label{hilb34:9}
  $9$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\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{hilb34:10}
  $10$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\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{hilb34:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\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{hilb34:12}
  $12$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb34:11]{$11$}
} \\
\label{hilb34:13}
  $13$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb34:12]{$12$}
} \\
\label{hilb34:14}
  $14$ & $(B\ \rightarrow \  P)\ \rightarrow \  (\neg P\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb34:13]{$13$}
} \\
\label{hilb34:15}
  $15$ & $((P\ \wedge \  P)\ \rightarrow \  P)\ \rightarrow \  (\neg P\ \rightarrow \  \neg (P\ \wedge \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  P$ in \hyperref[hilb34:14]{$14$}
} \\
\label{hilb34:16}
  $16$ & $\neg P\ \rightarrow \  \neg (P\ \wedge \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:8]{$8$}, \hyperref[hilb34:15]{$15$}} \\
\label{hilb34:17}
  $17$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\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{hilb34:18}
  $18$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb34:17]{$17$}
} \\
\label{hilb34:19}
  $19$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb34:18]{$18$}
} \\
\label{hilb34:20}
  $20$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb34:19]{$19$}
} \\
\label{hilb34:21}
  $21$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb34:20]{$20$}
} \\
\label{hilb34:22}
  $22$ & $(D\ \rightarrow \  \neg (P\ \wedge \  P))\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  P)$ in \hyperref[hilb34:21]{$21$}
} \\
\label{hilb34:23}
  $23$ & $(\neg P\ \rightarrow \  \neg (P\ \wedge \  P))\ \rightarrow \  ((Q\ \vee \  \neg P)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb34:22]{$22$}
} \\
\label{hilb34:24}
  $24$ & $(Q\ \vee \  \neg P)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:16]{$16$}, \hyperref[hilb34:23]{$23$}} \\
\label{hilb34:25}
  $25$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\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{hilb34:26}
  $26$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb34:25]{$25$}
} \\
\label{hilb34:27}
  $27$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb34:26]{$26$}
} \\
\label{hilb34:28}
  $28$ & $(B\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (P\ \wedge \  P)$ in \hyperref[hilb34:27]{$27$}
} \\
\label{hilb34:29}
  $29$ & $(Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb34:28]{$28$}
} \\
\label{hilb34:30}
  $30$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\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{hilb34:31}
  $31$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb34:30]{$30$}
} \\
\label{hilb34:32}
  $32$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb34:31]{$31$}
} \\
\label{hilb34:33}
  $33$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb34:32]{$32$}
} \\
\label{hilb34:34}
  $34$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \vee \  \neg P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg P)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  \neg P$ in \hyperref[hilb34:33]{$33$}
} \\
\label{hilb34:35}
  $35$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg P)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb34:34]{$34$}
} \\
\label{hilb34:36}
  $36$ & $((Q\ \vee \  \neg (P\ \wedge \  P))\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg P)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P)))\ \rightarrow \  ((Q\ \vee \  \neg P)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg (P\ \wedge \  P)$ in \hyperref[hilb34:35]{$35$}
} \\
\label{hilb34:37}
  $37$ & $((Q\ \vee \  \neg P)\ \rightarrow \  (Q\ \vee \  \neg (P\ \wedge \  P)))\ \rightarrow \  ((Q\ \vee \  \neg P)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:29]{$29$}, \hyperref[hilb34:36]{$36$}} \\
\label{hilb34:38}
  $38$ & $(Q\ \vee \  \neg P)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:24]{$24$}, \hyperref[hilb34:37]{$37$}} \\
\label{hilb34:39}
  $39$ & $(B\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb34:27]{$27$}
} \\
\label{hilb34:40}
  $40$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P$ in \hyperref[hilb34:39]{$39$}
} \\
\label{hilb34:41}
  $41$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg P\ \vee \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P\ \vee \  Q$ in \hyperref[hilb34:33]{$33$}
} \\
\label{hilb34:42}
  $42$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((\neg P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg P\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb34:41]{$41$}
} \\
\label{hilb34:43}
  $43$ & $((Q\ \vee \  \neg P)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((\neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg P))\ \rightarrow \  ((\neg P\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg P$ in \hyperref[hilb34:42]{$42$}
} \\
\label{hilb34:44}
  $44$ & $((\neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg P))\ \rightarrow \  ((\neg P\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:38]{$38$}, \hyperref[hilb34:43]{$43$}} \\
\label{hilb34:45}
  $45$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:40]{$40$}, \hyperref[hilb34:44]{$44$}} \\
\label{hilb34:46}
  $46$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb34:9]{$9$}
} \\
\label{hilb34:47}
  $47$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb34:46]{$46$}
} \\
\label{hilb34:48}
  $48$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  Q$ in \hyperref[hilb34:33]{$33$}
} \\
\label{hilb34:49}
  $49$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((P\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb34:48]{$48$}
} \\
\label{hilb34:50}
  $50$ & $((\neg P\ \vee \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  (((P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P\ \vee \  Q$ in \hyperref[hilb34:49]{$49$}
} \\
\label{hilb34:51}
  $51$ & $((P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:45]{$45$}, \hyperref[hilb34:50]{$50$}} \\
\label{hilb34:52}
  $52$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:9]{$9$}, \hyperref[hilb34:51]{$51$}} \\
\label{hilb34:53}
  $53$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb34:10]{$10$}
} \\
\label{hilb34:54}
  $54$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb34:53]{$53$}
} \\
\label{hilb34:55}
  $55$ & $(\neg B\ \vee \  Q)\ \rightarrow \  (B\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb34:54]{$54$}
} \\
\label{hilb34:56}
  $56$ & $(\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  P$ in \hyperref[hilb34:55]{$55$}
} \\
\label{hilb34:57}
  $57$ & $(D\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (((P\ \rightarrow \  Q)\ \rightarrow \  D)\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  P)\ \rightarrow \  Q$ in \hyperref[hilb34:48]{$48$}
} \\
\label{hilb34:58}
  $58$ & $((\neg (P\ \wedge \  P)\ \vee \  Q)\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (((P\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  P)\ \vee \  Q$ in \hyperref[hilb34:57]{$57$}
} \\
\label{hilb34:59}
  $59$ & $((P\ \rightarrow \  Q)\ \rightarrow \  (\neg (P\ \wedge \  P)\ \vee \  Q))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:56]{$56$}, \hyperref[hilb34:58]{$58$}} \\
\label{hilb34:60}
  $60$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:52]{$52$}, \hyperref[hilb34:59]{$59$}} \\
\label{hilb34:61}
  $61$ & $(B\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $(P\ \wedge \  P)\ \rightarrow \  Q$ in \hyperref[hilb34:13]{$13$}
} \\
\label{hilb34:62}
  $62$ & $((P\ \rightarrow \  Q)\ \rightarrow \  ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  \neg (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  Q$ in \hyperref[hilb34:61]{$61$}
} \\
\label{hilb34:63}
  $63$ & $\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  \neg (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:60]{$60$}, \hyperref[hilb34:62]{$62$}} \\
\label{hilb34:64}
  $64$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  D)\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb34:20]{$20$}
} \\
\label{hilb34:65}
  $65$ & $(D\ \rightarrow \  \neg (P\ \rightarrow \  Q))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  D)\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \rightarrow \  Q)$ in \hyperref[hilb34:64]{$64$}
} \\
\label{hilb34:66}
  $66$ & $(\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  \neg (P\ \rightarrow \  Q))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \wedge \  P)\ \rightarrow \  Q)$ in \hyperref[hilb34:65]{$65$}
} \\
\label{hilb34:67}
  $67$ & $((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:63]{$63$}, \hyperref[hilb34:66]{$66$}} \\
\label{hilb34:68}
  $68$ & $(B\ \vee \  \neg (P\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (P\ \rightarrow \  Q)$ in \hyperref[hilb34:27]{$27$}
} \\
\label{hilb34:69}
  $69$ & $((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb34:68]{$68$}
} \\
\label{hilb34:70}
  $70$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q)$ in \hyperref[hilb34:33]{$33$}
} \\
\label{hilb34:71}
  $71$ & $(D\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:70]{$70$}
} \\
\label{hilb34:72}
  $72$ & $(((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  ((((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q)))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q)$ in \hyperref[hilb34:71]{$71$}
} \\
\label{hilb34:73}
  $73$ & $(((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg (P\ \rightarrow \  Q)))\ \rightarrow \  (((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:69]{$69$}, \hyperref[hilb34:72]{$72$}} \\
\label{hilb34:74}
  $74$ & $((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:67]{$67$}, \hyperref[hilb34:73]{$73$}} \\
\label{hilb34:75}
  $75$ & $(B\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb34:27]{$27$}
} \\
\label{hilb34:76}
  $76$ & $(\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \wedge \  P)\ \rightarrow \  Q)$ in \hyperref[hilb34:75]{$75$}
} \\
\label{hilb34:77}
  $77$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:33]{$33$}
} \\
\label{hilb34:78}
  $78$ & $(D\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  (((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:77]{$77$}
} \\
\label{hilb34:79}
  $79$ & $(((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  (((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q)))\ \rightarrow \  ((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q)$ in \hyperref[hilb34:78]{$78$}
} \\
\label{hilb34:80}
  $80$ & $((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  (P\ \rightarrow \  Q))\ \vee \  \neg ((P\ \wedge \  P)\ \rightarrow \  Q)))\ \rightarrow \  ((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:74]{$74$}, \hyperref[hilb34:79]{$79$}} \\
\label{hilb34:81}
  $81$ & $(\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:76]{$76$}, \hyperref[hilb34:80]{$80$}} \\
\label{hilb34:82}
  $82$ & $(B\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg B\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb34:47]{$47$}
} \\
\label{hilb34:83}
  $83$ & $(((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \wedge \  P)\ \rightarrow \  Q$ in \hyperref[hilb34:82]{$82$}
} \\
\label{hilb34:84}
  $84$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  D)\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:33]{$33$}
} \\
\label{hilb34:85}
  $85$ & $(D\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  (((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  D)\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:84]{$84$}
} \\
\label{hilb34:86}
  $86$ & $((\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  (((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:85]{$85$}
} \\
\label{hilb34:87}
  $87$ & $((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg ((P\ \wedge \  P)\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:81]{$81$}, \hyperref[hilb34:86]{$86$}} \\
\label{hilb34:88}
  $88$ & $(((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:83]{$83$}, \hyperref[hilb34:87]{$87$}} \\
\label{hilb34:89}
  $89$ & $(\neg B\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (B\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \rightarrow \  (P\ \rightarrow \  Q)$ in \hyperref[hilb34:54]{$54$}
} \\
\label{hilb34:90}
  $90$ & $(\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \rightarrow \  Q$ in \hyperref[hilb34:89]{$89$}
} \\
\label{hilb34:91}
  $91$ & $(D\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  (((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  D)\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:84]{$84$}
} \\
\label{hilb34:92}
  $92$ & $((\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  (((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$ in \hyperref[hilb34:91]{$91$}
} \\
\label{hilb34:93}
  $93$ & $((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  (\neg (P\ \rightarrow \  Q)\ \vee \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))\ \rightarrow \  ((((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:90]{$90$}, \hyperref[hilb34:92]{$92$}} \\
\label{hilb34:94}
  $94$ & $(((P\ \wedge \  P)\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))\ \rightarrow \  ((P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:88]{$88$}, \hyperref[hilb34:93]{$93$}} \\
\label{hilb34:95}
  $95$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (P\ \rightarrow \  (P\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb34:7]{$7$}, \hyperref[hilb34:94]{$94$}} \\
 & & \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: & prophilbert3 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{prophilbert3_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{prophilbert3_1.00.00_1.00.00.pdf}  \\
\end{longtable}


\end{document}