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Theorem mercolem5 1506
 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
mercolem5 (θ → ((θφ) → (τ → (χφ))))

Proof of Theorem mercolem5
StepHypRef Expression
1 merco2 1501 . 2 (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))
2 merco2 1501 . . . . 5 (((φφ) → (( ⊥ → φ) → θ)) → ((θφ) → (τ → (χφ))))
3 mercolem1 1502 . . . . 5 ((((φφ) → (( ⊥ → φ) → θ)) → ((θφ) → (τ → (χφ)))) → ((( ⊥ → φ) → θ) → (θ → ((θφ) → (τ → (χφ))))))
42, 3ax-mp 8 . . . 4 ((( ⊥ → φ) → θ) → (θ → ((θφ) → (τ → (χφ)))))
5 mercolem2 1503 . . . . 5 (((θ → ((θφ) → (τ → (χφ)))) → θ) → (( ⊥ → φ) → (( ⊥ → φ) → θ)))
6 merco2 1501 . . . . 5 ((((θ → ((θφ) → (τ → (χφ)))) → θ) → (( ⊥ → φ) → (( ⊥ → φ) → θ))) → (((( ⊥ → φ) → θ) → (θ → ((θφ) → (τ → (χφ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (θ → ((θφ) → (τ → (χφ))))))))
75, 6ax-mp 8 . . . 4 (((( ⊥ → φ) → θ) → (θ → ((θφ) → (τ → (χφ))))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (θ → ((θφ) → (τ → (χφ)))))))
84, 7ax-mp 8 . . 3 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (θ → ((θφ) → (τ → (χφ))))))
91, 8ax-mp 8 . 2 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (θ → ((θφ) → (τ → (χφ)))))
101, 9ax-mp 8 1 (θ → ((θφ) → (τ → (χφ))))
 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:  mercolem6  1507  mercolem7  1508
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