MODULE (
  HEADER (
    SPEC (
      NAME ("predtheo2"),
      VERSION ("1.00.00"),
      VERSION ("1.00.00"),
      LOCATIONS (
        LOCATION ("http://www.qedeq.org/0_00_53")
      )
    ),
    HEADLINE ("Some more theorems of Predicate Calculus"),
    DESCRIPTION (
      "This module includes first proofs of predicate calculus theorems."
    ),
    EMAIL ("module@qedeq.org"),
    AUTHORS (
      AUTHOR ("Michael Meyling",EMAIL ("mime@qedeq.org"))
    )
  ),
  IMPORTS (
    IMPORT (
      SPEC (
        NAME ("predaxiom"),
        VERSION ("1.00.00"),
        VERSION ("1.00.00"),
        LOCATIONS (
          LOCATION ("."),
          LOCATION ("http://www.qedeq.org/0_00_53")
        )
      )
    ),
    IMPORT (
      SPEC (
        NAME ("prophilbert3"),
        VERSION ("1.00.00"),
        VERSION ("1.00.00"),
        LOCATIONS (
          LOCATION ("."),
          LOCATION ("http://www.qedeq.org/0_00_53")
        )
      )
    )
  ),
  PARAGRAPHS (
    PARAGRAPH (
      LABEL ("predtheo1"),
      "A simple implication:",
      PROPOSITION (
        SENTENCE (
          IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))
        ),
        PROOF (
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(2)))),
            ADDAXIOM ( LINK ("axiom5"))
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(2))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            ADDAXIOM ( LINK ("axiom6"))
          ),
          LINE (
            IMPL(IMPL(PROP(1), PROP(2)), IMPL(IMPL(PROP(3), PROP(1)), IMPL(PROP(3), PROP(2)))),
            ADDSENTENCE ( LINK ("hilb1"))
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(2)), IMPL(IMPL(PROP(3), PROP(4)), IMPL(PROP(3), PROP(2)))),
            REPLACEPROP (
              3, 
              PROP(1),
              PROP(4)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(5)), IMPL(IMPL(PROP(3), PROP(4)), IMPL(PROP(3), PROP(5)))),
            REPLACEPROP (
              4, 
              PROP(2),
              PROP(5)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(5)), IMPL(IMPL(PROP(6), PROP(4)), IMPL(PROP(6), PROP(5)))),
            REPLACEPROP (
              5, 
              PROP(3),
              PROP(6)
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(2))), PROP(5)), IMPL(IMPL(PROP(6), PREDVAR(1, L(VAR(2)))), IMPL(PROP(6), PROP(5)))),
            REPLACEPROP (
              6, 
              PROP(4),
              PREDVAR(1, L(VAR(2)))
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(2))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(IMPL(PROP(6), PREDVAR(1, L(VAR(2)))), IMPL(PROP(6), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))),
            REPLACEPROP (
              7, 
              PROP(5),
              EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(2))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(2)))), IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))),
            REPLACEPROP (
              8, 
              PROP(6),
              FORALL(VAR(1), PREDVAR(1, L(VAR(1))))
            )
          ),
          LINE (
            IMPL(IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(2)))), IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))),
            MODUSPONENS (2, 9)
          ),
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            MODUSPONENS (1, 10)
          )
        )
      )
    ),
    PARAGRAPH (
      LABEL ("predtheo2"),
      "A well known implication:",
      PROPOSITION (
        SENTENCE (
          IMPL(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))), NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))))
        ),
        PROOF (
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(2)))),
            ADDAXIOM ( LINK ("axiom5"))
          ),
          LINE (
            IMPL(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1))))), NOT(PREDVAR(1, L(VAR(2))))),
            REPLACEPREDICATE (
              1, 
              PREDVAR(1, L(SPATTERN(1))),
              NOT(PREDVAR(1, L(SPATTERN(1))))
            )
          ),
          LINE (
            OR(NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))), NOT(PREDVAR(1, L(VAR(2))))),
            USEABBREVIATION (2, LINK ("impl"), 1)
          ),
          LINE (
            IMPL(OR(PROP(1), PROP(2)), OR(PROP(2), PROP(1))),
            ADDAXIOM ( LINK ("axiom3"))
          ),
          LINE (
            IMPL(OR(NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))), PROP(2)), OR(PROP(2), NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              4, 
              PROP(1),
              NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1))))))
            )
          ),
          LINE (
            IMPL(OR(NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))), NOT(PREDVAR(1, L(VAR(2))))), OR(NOT(PREDVAR(1, L(VAR(2)))), NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              5, 
              PROP(2),
              NOT(PREDVAR(1, L(VAR(2))))
            )
          ),
          LINE (
            OR(NOT(PREDVAR(1, L(VAR(2)))), NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1))))))),
            MODUSPONENS (3, 6)
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(2))), NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1))))))),
            REVERSEABBREVIATION (7, LINK ("impl"), 1)
          ),
          LINE (
            IMPL(EXISTS(VAR(2), PREDVAR(1, L(VAR(2)))), NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1))))))),
            PARTICULARIZATION (8, VAR(2))
          ),
          LINE (
            IMPL(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))), NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1))))))),
            RENAMEBOUNDVARIABLE (9, VAR(2), VAR(1), 1)
          )
        )
      )
    ),
    PARAGRAPH (
      LABEL ("predtheo3"),
      "The reverse is also true:",
      PROPOSITION (
        SENTENCE (
          IMPL(NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))
        ),
        PROOF (
          LINE (
            IMPL(PREDVAR(1, L(VAR(2))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            ADDAXIOM ( LINK ("axiom6"))
          ),
          LINE (
            IMPL(IMPL(PROP(1), PROP(2)), IMPL(NOT(PROP(2)), NOT(PROP(1)))),
            ADDSENTENCE ( LINK ("hilb7"))
          ),
          LINE (
            IMPL(IMPL(PROP(3), PROP(2)), IMPL(NOT(PROP(2)), NOT(PROP(3)))),
            REPLACEPROP (
              2, 
              PROP(1),
              PROP(3)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(3), PROP(4)), IMPL(NOT(PROP(4)), NOT(PROP(3)))),
            REPLACEPROP (
              3, 
              PROP(2),
              PROP(4)
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(2))), PROP(4)), IMPL(NOT(PROP(4)), NOT(PREDVAR(1, L(VAR(2)))))),
            REPLACEPROP (
              4, 
              PROP(3),
              PREDVAR(1, L(VAR(2)))
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(2))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), NOT(PREDVAR(1, L(VAR(2)))))),
            REPLACEPROP (
              5, 
              PROP(4),
              EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))
            )
          ),
          LINE (
            IMPL(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), NOT(PREDVAR(1, L(VAR(2))))),
            MODUSPONENS (1, 6)
          ),
          LINE (
            IMPL(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))),
            GENERALIZATION (7, VAR(2))
          ),
          LINE (
            IMPL(IMPL(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), PROP(4)), IMPL(NOT(PROP(4)), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              4, 
              PROP(3),
              NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))
            )
          ),
          LINE (
            IMPL(IMPL(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              9, 
              PROP(4),
              FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))
            )
          ),
          LINE (
            IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))),
            MODUSPONENS (8, 10)
          ),
          LINE (
            IMPL(NOT(NOT(PROP(1))), PROP(1)),
            ADDSENTENCE ( LINK ("hilb6"))
          ),
          LINE (
            IMPL(NOT(NOT(PROP(2))), PROP(2)),
            REPLACEPROP (
              12, 
              PROP(1),
              PROP(2)
            )
          ),
          LINE (
            IMPL(NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            REPLACEPROP (
              13, 
              PROP(2),
              EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))
            )
          ),
          LINE (
            IMPL(IMPL(PROP(1), PROP(2)), OR(NOT(PROP(1)), PROP(2))),
            ADDSENTENCE ( LINK ("defimpl1"))
          ),
          LINE (
            IMPL(OR(NOT(PROP(1)), PROP(2)), IMPL(PROP(1), PROP(2))),
            ADDSENTENCE ( LINK ("defimpl2"))
          ),
          LINE (
            IMPL(IMPL(PROP(1), PROP(2)), IMPL(OR(PROP(3), PROP(1)), OR(PROP(3), PROP(2)))),
            ADDAXIOM ( LINK ("axiom4"))
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(2)), IMPL(OR(PROP(3), PROP(4)), OR(PROP(3), PROP(2)))),
            REPLACEPROP (
              17, 
              PROP(1),
              PROP(4)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(5)), IMPL(OR(PROP(3), PROP(4)), OR(PROP(3), PROP(5)))),
            REPLACEPROP (
              18, 
              PROP(2),
              PROP(5)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(5)), IMPL(OR(PROP(6), PROP(4)), OR(PROP(6), PROP(5)))),
            REPLACEPROP (
              19, 
              PROP(3),
              PROP(6)
            )
          ),
          LINE (
            IMPL(IMPL(NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), PROP(5)), IMPL(OR(PROP(6), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(PROP(6), PROP(5)))),
            REPLACEPROP (
              20, 
              PROP(4),
              NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))
            )
          ),
          LINE (
            IMPL(IMPL(NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(OR(PROP(6), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(PROP(6), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))),
            REPLACEPROP (
              21, 
              PROP(5),
              EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))
            )
          ),
          LINE (
            IMPL(IMPL(NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))),
            REPLACEPROP (
              22, 
              PROP(6),
              NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))))
            )
          ),
          LINE (
            IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))),
            MODUSPONENS (14, 23)
          ),
          LINE (
            IMPL(IMPL(PROP(3), PROP(2)), OR(NOT(PROP(3)), PROP(2))),
            REPLACEPROP (
              15, 
              PROP(1),
              PROP(3)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(3), PROP(4)), OR(NOT(PROP(3)), PROP(4))),
            REPLACEPROP (
              25, 
              PROP(2),
              PROP(4)
            )
          ),
          LINE (
            IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), PROP(4)), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), PROP(4))),
            REPLACEPROP (
              26, 
              PROP(3),
              NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))
            )
          ),
          LINE (
            IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              27, 
              PROP(4),
              NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))
            )
          ),
          LINE (
            IMPL(IMPL(PROP(1), PROP(2)), IMPL(IMPL(PROP(3), PROP(1)), IMPL(PROP(3), PROP(2)))),
            ADDSENTENCE ( LINK ("hilb1"))
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(2)), IMPL(IMPL(PROP(3), PROP(4)), IMPL(PROP(3), PROP(2)))),
            REPLACEPROP (
              29, 
              PROP(1),
              PROP(4)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(5)), IMPL(IMPL(PROP(3), PROP(4)), IMPL(PROP(3), PROP(5)))),
            REPLACEPROP (
              30, 
              PROP(2),
              PROP(5)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(4), PROP(5)), IMPL(IMPL(PROP(6), PROP(4)), IMPL(PROP(6), PROP(5)))),
            REPLACEPROP (
              31, 
              PROP(3),
              PROP(6)
            )
          ),
          LINE (
            IMPL(IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), PROP(5)), IMPL(IMPL(PROP(6), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))), IMPL(PROP(6), PROP(5)))),
            REPLACEPROP (
              32, 
              PROP(4),
              OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))
            )
          ),
          LINE (
            IMPL(IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(IMPL(PROP(6), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))), IMPL(PROP(6), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              33, 
              PROP(5),
              OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))
            )
          ),
          LINE (
            IMPL(IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))), IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              34, 
              PROP(6),
              IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))
            )
          ),
          LINE (
            IMPL(IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))), IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))),
            MODUSPONENS (24, 35)
          ),
          LINE (
            IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))),
            MODUSPONENS (28, 36)
          ),
          LINE (
            IMPL(OR(NOT(PROP(3)), PROP(2)), IMPL(PROP(3), PROP(2))),
            REPLACEPROP (
              16, 
              PROP(1),
              PROP(3)
            )
          ),
          LINE (
            IMPL(OR(NOT(PROP(3)), PROP(4)), IMPL(PROP(3), PROP(4))),
            REPLACEPROP (
              38, 
              PROP(2),
              PROP(4)
            )
          ),
          LINE (
            IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), PROP(4)), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), PROP(4))),
            REPLACEPROP (
              39, 
              PROP(3),
              NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))
            )
          ),
          LINE (
            IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))),
            REPLACEPROP (
              40, 
              PROP(4),
              EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))
            )
          ),
          LINE (
            IMPL(IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), PROP(5)), IMPL(IMPL(PROP(6), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(PROP(6), PROP(5)))),
            REPLACEPROP (
              32, 
              PROP(4),
              OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))
            )
          ),
          LINE (
            IMPL(IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(IMPL(PROP(6), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(PROP(6), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              42, 
              PROP(5),
              IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))
            )
          ),
          LINE (
            IMPL(IMPL(OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))),
            REPLACEPROP (
              43, 
              PROP(6),
              IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))))
            )
          ),
          LINE (
            IMPL(IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), OR(NOT(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2))))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))), IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))),
            MODUSPONENS (41, 44)
          ),
          LINE (
            IMPL(IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), NOT(NOT(EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))))), IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1)))))),
            MODUSPONENS (37, 45)
          ),
          LINE (
            IMPL(NOT(FORALL(VAR(2), NOT(PREDVAR(1, L(VAR(2)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            MODUSPONENS (11, 46)
          ),
          LINE (
            IMPL(NOT(FORALL(VAR(1), NOT(PREDVAR(1, L(VAR(1)))))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            RENAMEBOUNDVARIABLE (47, VAR(2), VAR(1), 1)
          )
        )
      )
    ),
    PARAGRAPH (
      LABEL ("predtheo4"),
      "Exchange of universal quantors:",
      PROPOSITION (
        SENTENCE (
          IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), FORALL(VAR(1), PREDVAR(1, L(VAR(1), VAR(2))))))
        ),
        PROOF (
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(2)))),
            ADDAXIOM ( LINK ("axiom5"))
          ),
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(6)))),
            RENAMEFREEVARIABLE (1, VAR(2), VAR(6))
          ),
          LINE (
            IMPL(FORALL(VAR(7), PREDVAR(1, L(VAR(7)))), PREDVAR(1, L(VAR(6)))),
            RENAMEBOUNDVARIABLE (2, VAR(1), VAR(7), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(7), PREDVAR(1, L(VAR(7)))), PREDVAR(1, L(VAR(4)))),
            RENAMEFREEVARIABLE (3, VAR(6), VAR(4))
          ),
          LINE (
            IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(2)))), PREDVAR(1, L(VAR(4)))),
            RENAMEBOUNDVARIABLE (4, VAR(7), VAR(2), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2)))), PREDVAR(1, L(VAR(3), VAR(4)))),
            REPLACEPREDICATE (
              5, 
              PREDVAR(1, L(SPATTERN(1))),
              PREDVAR(1, L(VAR(3), SPATTERN(1)))
            )
          ),
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(7)))),
            RENAMEFREEVARIABLE (1, VAR(2), VAR(7))
          ),
          LINE (
            IMPL(FORALL(VAR(8), PREDVAR(1, L(VAR(8)))), PREDVAR(1, L(VAR(7)))),
            RENAMEBOUNDVARIABLE (7, VAR(1), VAR(8), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(8), PREDVAR(1, L(VAR(8)))), PREDVAR(1, L(VAR(3)))),
            RENAMEFREEVARIABLE (8, VAR(7), VAR(3))
          ),
          LINE (
            IMPL(FORALL(VAR(5), PREDVAR(1, L(VAR(5)))), PREDVAR(1, L(VAR(3)))),
            RENAMEBOUNDVARIABLE (9, VAR(8), VAR(5), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(5), FORALL(VAR(6), PREDVAR(1, L(VAR(5), VAR(6))))), FORALL(VAR(6), PREDVAR(1, L(VAR(3), VAR(6))))),
            REPLACEPREDICATE (
              10, 
              PREDVAR(1, L(SPATTERN(1))),
              FORALL(VAR(6), PREDVAR(1, L(SPATTERN(1), VAR(6))))
            )
          ),
          LINE (
            IMPL(FORALL(VAR(12), FORALL(VAR(6), PREDVAR(1, L(VAR(12), VAR(6))))), FORALL(VAR(6), PREDVAR(1, L(VAR(3), VAR(6))))),
            RENAMEBOUNDVARIABLE (11, VAR(5), VAR(12), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(12), FORALL(VAR(13), PREDVAR(1, L(VAR(12), VAR(13))))), FORALL(VAR(6), PREDVAR(1, L(VAR(3), VAR(6))))),
            RENAMEBOUNDVARIABLE (12, VAR(6), VAR(13), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(12), FORALL(VAR(13), PREDVAR(1, L(VAR(12), VAR(13))))), FORALL(VAR(14), PREDVAR(1, L(VAR(3), VAR(14))))),
            RENAMEBOUNDVARIABLE (13, VAR(6), VAR(14), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(13), PREDVAR(1, L(VAR(1), VAR(13))))), FORALL(VAR(14), PREDVAR(1, L(VAR(3), VAR(14))))),
            RENAMEBOUNDVARIABLE (14, VAR(12), VAR(1), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(14), PREDVAR(1, L(VAR(3), VAR(14))))),
            RENAMEBOUNDVARIABLE (15, VAR(13), VAR(2), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2))))),
            RENAMEBOUNDVARIABLE (16, VAR(14), VAR(2), 1)
          ),
          LINE (
            IMPL(IMPL(PROP(1), PROP(2)), IMPL(IMPL(PROP(3), PROP(1)), IMPL(PROP(3), PROP(2)))),
            ADDSENTENCE ( LINK ("hilb1"))
          ),
          LINE (
            IMPL(IMPL(PROP(5), PROP(2)), IMPL(IMPL(PROP(3), PROP(5)), IMPL(PROP(3), PROP(2)))),
            REPLACEPROP (
              18, 
              PROP(1),
              PROP(5)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(5), PROP(6)), IMPL(IMPL(PROP(3), PROP(5)), IMPL(PROP(3), PROP(6)))),
            REPLACEPROP (
              19, 
              PROP(2),
              PROP(6)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(5), PROP(6)), IMPL(IMPL(PROP(7), PROP(5)), IMPL(PROP(7), PROP(6)))),
            REPLACEPROP (
              20, 
              PROP(3),
              PROP(7)
            )
          ),
          LINE (
            IMPL(IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2)))), PROP(6)), IMPL(IMPL(PROP(7), FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2))))), IMPL(PROP(7), PROP(6)))),
            REPLACEPROP (
              21, 
              PROP(5),
              FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2))))
            )
          ),
          LINE (
            IMPL(IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2)))), PREDVAR(1, L(VAR(3), VAR(4)))), IMPL(IMPL(PROP(7), FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2))))), IMPL(PROP(7), PREDVAR(1, L(VAR(3), VAR(4)))))),
            REPLACEPROP (
              22, 
              PROP(6),
              PREDVAR(1, L(VAR(3), VAR(4)))
            )
          ),
          LINE (
            IMPL(IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2)))), PREDVAR(1, L(VAR(3), VAR(4)))), IMPL(IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2))))), IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), PREDVAR(1, L(VAR(3), VAR(4)))))),
            REPLACEPROP (
              23, 
              PROP(7),
              FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2)))))
            )
          ),
          LINE (
            IMPL(IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), PREDVAR(1, L(VAR(3), VAR(2))))), IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), PREDVAR(1, L(VAR(3), VAR(4))))),
            MODUSPONENS (6, 24)
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), PREDVAR(1, L(VAR(3), VAR(4)))),
            MODUSPONENS (17, 25)
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))),
            GENERALIZATION (26, VAR(3))
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(4), FORALL(VAR(3), PREDVAR(1, L(VAR(3), VAR(4)))))),
            GENERALIZATION (27, VAR(4))
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), FORALL(VAR(3), PREDVAR(1, L(VAR(3), VAR(2)))))),
            RENAMEBOUNDVARIABLE (28, VAR(4), VAR(2), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), FORALL(VAR(1), PREDVAR(1, L(VAR(1), VAR(2)))))),
            RENAMEBOUNDVARIABLE (29, VAR(3), VAR(1), 1)
          )
        )
      )
    ),
    PARAGRAPH (
      LABEL ("predtheo5"),
      "Implication of changing sequence of existence and universal quantor:",
      PROPOSITION (
        SENTENCE (
          IMPL(EXISTS(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), EXISTS(VAR(1), PREDVAR(1, L(VAR(1), VAR(2))))))
        ),
        PROOF (
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(2)))),
            ADDAXIOM ( LINK ("axiom5"))
          ),
          LINE (
            IMPL(FORALL(VAR(1), PREDVAR(1, L(VAR(1)))), PREDVAR(1, L(VAR(6)))),
            RENAMEFREEVARIABLE (1, VAR(2), VAR(6))
          ),
          LINE (
            IMPL(FORALL(VAR(7), PREDVAR(1, L(VAR(7)))), PREDVAR(1, L(VAR(6)))),
            RENAMEBOUNDVARIABLE (2, VAR(1), VAR(7), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(7), PREDVAR(1, L(VAR(7)))), PREDVAR(1, L(VAR(4)))),
            RENAMEFREEVARIABLE (3, VAR(6), VAR(4))
          ),
          LINE (
            IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(2)))), PREDVAR(1, L(VAR(4)))),
            RENAMEBOUNDVARIABLE (4, VAR(7), VAR(2), 1)
          ),
          LINE (
            IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2)))), PREDVAR(1, L(VAR(1), VAR(4)))),
            REPLACEPREDICATE (
              5, 
              PREDVAR(1, L(SPATTERN(1))),
              PREDVAR(1, L(VAR(1), SPATTERN(1)))
            )
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(2))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            ADDAXIOM ( LINK ("axiom6"))
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(5))), EXISTS(VAR(1), PREDVAR(1, L(VAR(1))))),
            RENAMEFREEVARIABLE (7, VAR(2), VAR(5))
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(5))), EXISTS(VAR(6), PREDVAR(1, L(VAR(6))))),
            RENAMEBOUNDVARIABLE (8, VAR(1), VAR(6), 1)
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(1))), EXISTS(VAR(6), PREDVAR(1, L(VAR(6))))),
            RENAMEFREEVARIABLE (9, VAR(5), VAR(1))
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(1))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3))))),
            RENAMEBOUNDVARIABLE (10, VAR(6), VAR(3), 1)
          ),
          LINE (
            IMPL(PREDVAR(1, L(VAR(1), VAR(4))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))),
            REPLACEPREDICATE (
              11, 
              PREDVAR(1, L(SPATTERN(1))),
              PREDVAR(1, L(SPATTERN(1), VAR(4)))
            )
          ),
          LINE (
            IMPL(IMPL(PROP(1), PROP(2)), IMPL(IMPL(PROP(3), PROP(1)), IMPL(PROP(3), PROP(2)))),
            ADDSENTENCE ( LINK ("hilb1"))
          ),
          LINE (
            IMPL(IMPL(PROP(5), PROP(2)), IMPL(IMPL(PROP(3), PROP(5)), IMPL(PROP(3), PROP(2)))),
            REPLACEPROP (
              13, 
              PROP(1),
              PROP(5)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(5), PROP(6)), IMPL(IMPL(PROP(3), PROP(5)), IMPL(PROP(3), PROP(6)))),
            REPLACEPROP (
              14, 
              PROP(2),
              PROP(6)
            )
          ),
          LINE (
            IMPL(IMPL(PROP(5), PROP(6)), IMPL(IMPL(PROP(7), PROP(5)), IMPL(PROP(7), PROP(6)))),
            REPLACEPROP (
              15, 
              PROP(3),
              PROP(7)
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(1), VAR(4))), PROP(6)), IMPL(IMPL(PROP(7), PREDVAR(1, L(VAR(1), VAR(4)))), IMPL(PROP(7), PROP(6)))),
            REPLACEPROP (
              16, 
              PROP(5),
              PREDVAR(1, L(VAR(1), VAR(4)))
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(1), VAR(4))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))), IMPL(IMPL(PROP(7), PREDVAR(1, L(VAR(1), VAR(4)))), IMPL(PROP(7), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))))),
            REPLACEPROP (
              17, 
              PROP(6),
              EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))
            )
          ),
          LINE (
            IMPL(IMPL(PREDVAR(1, L(VAR(1), VAR(4))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))), IMPL(IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2)))), PREDVAR(1, L(VAR(1), VAR(4)))), IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2)))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))))),
            REPLACEPROP (
              18, 
              PROP(7),
              FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))
            )
          ),
          LINE (
            IMPL(IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2)))), PREDVAR(1, L(VAR(1), VAR(4)))), IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2)))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4)))))),
            MODUSPONENS (12, 19)
          ),
          LINE (
            IMPL(FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2)))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))),
            MODUSPONENS (6, 20)
          ),
          LINE (
            IMPL(EXISTS(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4))))),
            PARTICULARIZATION (21, VAR(1))
          ),
          LINE (
            IMPL(EXISTS(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(4), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(4)))))),
            GENERALIZATION (22, VAR(4))
          ),
          LINE (
            IMPL(EXISTS(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(12), EXISTS(VAR(3), PREDVAR(1, L(VAR(3), VAR(12)))))),
            RENAMEBOUNDVARIABLE (23, VAR(4), VAR(12), 1)
          ),
          LINE (
            IMPL(EXISTS(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(12), EXISTS(VAR(13), PREDVAR(1, L(VAR(13), VAR(12)))))),
            RENAMEBOUNDVARIABLE (24, VAR(3), VAR(13), 1)
          ),
          LINE (
            IMPL(EXISTS(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), EXISTS(VAR(13), PREDVAR(1, L(VAR(13), VAR(2)))))),
            RENAMEBOUNDVARIABLE (25, VAR(12), VAR(2), 1)
          ),
          LINE (
            IMPL(EXISTS(VAR(1), FORALL(VAR(2), PREDVAR(1, L(VAR(1), VAR(2))))), FORALL(VAR(2), EXISTS(VAR(1), PREDVAR(1, L(VAR(1), VAR(2)))))),
            RENAMEBOUNDVARIABLE (26, VAR(13), VAR(1), 1)
          )
        )
      )
    )
  )
)

