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Theorem merco1lem3 1483
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem3 (((φψ) → (χ → ⊥ )) → (χφ))

Proof of Theorem merco1lem3
StepHypRef Expression
1 merco1lem2 1482 . . 3 (((φφ) → ⊥ ) → (((φφ) → (φ → ⊥ )) → ⊥ ))
2 retbwax2 1481 . . . 4 ((((φφ) → (φ → ⊥ )) → (φφ)) → (φ → (((φφ) → (φ → ⊥ )) → (φφ))))
3 merco1lem2 1482 . . . 4 (((((φφ) → (φ → ⊥ )) → (φφ)) → (φ → (((φφ) → (φ → ⊥ )) → (φφ)))) → ((((φφ) → ⊥ ) → (((φφ) → (φ → ⊥ )) → ⊥ )) → (φ → (((φφ) → (φ → ⊥ )) → (φφ)))))
42, 3ax-mp 5 . . 3 ((((φφ) → ⊥ ) → (((φφ) → (φ → ⊥ )) → ⊥ )) → (φ → (((φφ) → (φ → ⊥ )) → (φφ))))
51, 4ax-mp 5 . 2 (φ → (((φφ) → (φ → ⊥ )) → (φφ)))
6 merco1lem2 1482 . . 3 (((χφ) → ⊥ ) → (((φψ) → (χ → ⊥ )) → ⊥ ))
7 retbwax2 1481 . . . 4 ((((φψ) → (χ → ⊥ )) → (χφ)) → ((φ → (((φφ) → (φ → ⊥ )) → (φφ))) → (((φψ) → (χ → ⊥ )) → (χφ))))
8 merco1lem2 1482 . . . 4 (((((φψ) → (χ → ⊥ )) → (χφ)) → ((φ → (((φφ) → (φ → ⊥ )) → (φφ))) → (((φψ) → (χ → ⊥ )) → (χφ)))) → ((((χφ) → ⊥ ) → (((φψ) → (χ → ⊥ )) → ⊥ )) → ((φ → (((φφ) → (φ → ⊥ )) → (φφ))) → (((φψ) → (χ → ⊥ )) → (χφ)))))
97, 8ax-mp 5 . . 3 ((((χφ) → ⊥ ) → (((φψ) → (χ → ⊥ )) → ⊥ )) → ((φ → (((φφ) → (φ → ⊥ )) → (φφ))) → (((φψ) → (χ → ⊥ )) → (χφ))))
106, 9ax-mp 5 . 2 ((φ → (((φφ) → (φ → ⊥ )) → (φφ))) → (((φψ) → (χ → ⊥ )) → (χφ)))
115, 10ax-mp 5 1 (((φψ) → (χ → ⊥ )) → (χφ))
 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:  merco1lem4  1484  merco1lem6  1486  merco1lem11  1492  merco1lem12  1493  merco1lem18  1499
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