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Theorem mercolem6 1507
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem6 ((φ → (ψ → (φχ))) → (ψ → (φχ)))

Proof of Theorem mercolem6
StepHypRef Expression
1 merco2 1501 . 2 (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))
2 mercolem1 1502 . . . . . . . 8 (((φ → (φ → (ψ → (φχ)))) → (φχ)) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))
3 mercolem1 1502 . . . . . . . 8 ((((φ → (φ → (ψ → (φχ)))) → (φχ)) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))) → ((φχ) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))
42, 3ax-mp 8 . . . . . . 7 ((φχ) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))
5 mercolem5 1506 . . . . . . . 8 (φ → ((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))
6 mercolem4 1505 . . . . . . . 8 ((φ → ((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))) → (((φχ) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))))
75, 6ax-mp 8 . . . . . . 7 (((φχ) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))))
84, 7ax-mp 8 . . . . . 6 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))
91, 8ax-mp 8 . . . . 5 ((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))
10 mercolem1 1502 . . . . . . . 8 (((φ → (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))
11 mercolem1 1502 . . . . . . . 8 ((((φ → (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))) → (((φ → (ψ → (φχ))) → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))))
1210, 11ax-mp 8 . . . . . . 7 (((φ → (ψ → (φχ))) → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))))
13 mercolem5 1506 . . . . . . . 8 ((φ → (ψ → (φχ))) → (((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))))
14 mercolem4 1505 . . . . . . . 8 (((φ → (ψ → (φχ))) → (((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))))) → ((((φ → (ψ → (φχ))) → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))))))
1513, 14ax-mp 8 . . . . . . 7 ((((φ → (ψ → (φχ))) → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))))))
1612, 15ax-mp 8 . . . . . 6 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))))
171, 16ax-mp 8 . . . . 5 (((φ → (ψ → (φχ))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))))
189, 17ax-mp 8 . . . 4 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))))
191, 18ax-mp 8 . . 3 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ)))))
201, 19ax-mp 8 . 2 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((φ → (ψ → (φχ))) → (ψ → (φχ))))
211, 20ax-mp 8 1 ((φ → (ψ → (φχ))) → (ψ → (φχ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊥ wfal 1317 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320 This theorem is referenced by:  mercolem7  1508  re1tbw1  1510  re1tbw2  1511  re1tbw3  1512
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