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Theorem retbwax2 1481
 Description: tbw-ax2 1466 rederived from merco1 1478. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
retbwax2 (φ → (ψφ))

Proof of Theorem retbwax2
StepHypRef Expression
1 merco1lem1 1479 . . . 4 (((((φφ) → φ) → (φ → ⊥ )) → φ) → ( ⊥ → φ))
2 merco1 1478 . . . 4 ((((((φφ) → φ) → (φ → ⊥ )) → φ) → ( ⊥ → φ)) → ((( ⊥ → φ) → (φφ)) → (φ → (φφ))))
31, 2ax-mp 5 . . 3 ((( ⊥ → φ) → (φφ)) → (φ → (φφ)))
4 merco1 1478 . . . 4 (((((φ → (φφ)) → (φ → ⊥ )) → (φ → ⊥ )) → ⊥ ) → (( ⊥ → φ) → (φφ)))
5 merco1 1478 . . . 4 ((((((φ → (φφ)) → (φ → ⊥ )) → (φ → ⊥ )) → ⊥ ) → (( ⊥ → φ) → (φφ))) → (((( ⊥ → φ) → (φφ)) → (φ → (φφ))) → (φ → (φ → (φφ)))))
64, 5ax-mp 5 . . 3 (((( ⊥ → φ) → (φφ)) → (φ → (φφ))) → (φ → (φ → (φφ))))
73, 6ax-mp 5 . 2 (φ → (φ → (φφ)))
8 merco1lem1 1479 . . . 4 (((((ψφ) → φ) → (φ → ⊥ )) → φ) → ( ⊥ → φ))
9 merco1 1478 . . . 4 ((((((ψφ) → φ) → (φ → ⊥ )) → φ) → ( ⊥ → φ)) → ((( ⊥ → φ) → (ψφ)) → (φ → (ψφ))))
108, 9ax-mp 5 . . 3 ((( ⊥ → φ) → (ψφ)) → (φ → (ψφ)))
11 merco1 1478 . . . 4 (((((φ → (ψφ)) → (ψ → ⊥ )) → ((φ → (φ → (φφ))) → ⊥ )) → ⊥ ) → (( ⊥ → φ) → (ψφ)))
12 merco1 1478 . . . 4 ((((((φ → (ψφ)) → (ψ → ⊥ )) → ((φ → (φ → (φφ))) → ⊥ )) → ⊥ ) → (( ⊥ → φ) → (ψφ))) → (((( ⊥ → φ) → (ψφ)) → (φ → (ψφ))) → ((φ → (φ → (φφ))) → (φ → (ψφ)))))
1311, 12ax-mp 5 . . 3 (((( ⊥ → φ) → (ψφ)) → (φ → (ψφ))) → ((φ → (φ → (φφ))) → (φ → (ψφ))))
1410, 13ax-mp 5 . 2 ((φ → (φ → (φφ))) → (φ → (ψφ)))
157, 14ax-mp 5 1 (φ → (ψφ))
 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:  merco1lem2  1482  merco1lem3  1483  retbwax3  1488
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