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Theorem mercolem4 1505
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
mercolem4 ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))

Proof of Theorem mercolem4
StepHypRef Expression
1 merco2 1501 . 2 (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))
2 merco2 1501 . . . 4 ((((ηφ) → φ) → (( ⊥ → φ) → θ)) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))))
3 merco2 1501 . . . . . . . . 9 (((φφ) → (( ⊥ → φ) → (θχ))) → (((θχ) → φ) → (τ → (ηφ))))
4 mercolem1 1502 . . . . . . . . 9 ((((φφ) → (( ⊥ → φ) → (θχ))) → (((θχ) → φ) → (τ → (ηφ)))) → ((( ⊥ → φ) → (θχ)) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))))
53, 4ax-mp 5 . . . . . . . 8 ((( ⊥ → φ) → (θχ)) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))))
6 mercolem1 1502 . . . . . . . 8 (((( ⊥ → φ) → (θχ)) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))) → ((θχ) → (( ⊥ → φ) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))))))
75, 6ax-mp 5 . . . . . . 7 ((θχ) → (( ⊥ → φ) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))))
8 merco2 1501 . . . . . . 7 (((θχ) → (( ⊥ → φ) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))))) → ((((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))) → θ) → (((ηφ) → φ) → (( ⊥ → φ) → θ))))
97, 8ax-mp 5 . . . . . 6 ((((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))) → θ) → (((ηφ) → φ) → (( ⊥ → φ) → θ)))
10 mercolem3 1504 . . . . . 6 (((((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))) → θ) → (((ηφ) → φ) → (( ⊥ → φ) → θ))) → ((((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))) → θ) → (( ⊥ → φ) → (((ηφ) → φ) → (( ⊥ → φ) → θ)))))
119, 10ax-mp 5 . . . . 5 ((((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))) → θ) → (( ⊥ → φ) → (((ηφ) → φ) → (( ⊥ → φ) → θ))))
12 merco2 1501 . . . . 5 (((((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))) → θ) → (( ⊥ → φ) → (((ηφ) → φ) → (( ⊥ → φ) → θ)))) → (((((ηφ) → φ) → (( ⊥ → φ) → θ)) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))))))
1311, 12ax-mp 5 . . . 4 (((((ηφ) → φ) → (( ⊥ → φ) → θ)) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))))))
142, 13ax-mp 5 . . 3 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))))
151, 14ax-mp 5 . 2 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ)))))
161, 15ax-mp 5 1 ((θ → (ηφ)) → (((θχ) → φ) → (τ → (ηφ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320
This theorem is referenced by:  mercolem6  1507  mercolem7  1508
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