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Theorem merco1lem12 1493
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem12 ((φψ) → (((χ → (φτ)) → φ) → ψ))

Proof of Theorem merco1lem12
StepHypRef Expression
1 merco1lem3 1483 . . . . 5 ((((φτ) → (((χ → (φτ)) → φ) → ⊥ )) → (χ → ⊥ )) → (χ → (φτ)))
2 merco1 1478 . . . . 5 (((((φτ) → (((χ → (φτ)) → φ) → ⊥ )) → (χ → ⊥ )) → (χ → (φτ))) → (((χ → (φτ)) → φ) → (((χ → (φτ)) → φ) → φ)))
31, 2ax-mp 5 . . . 4 (((χ → (φτ)) → φ) → (((χ → (φτ)) → φ) → φ))
4 merco1lem9 1490 . . . 4 ((((χ → (φτ)) → φ) → (((χ → (φτ)) → φ) → φ)) → (((χ → (φτ)) → φ) → φ))
53, 4ax-mp 5 . . 3 (((χ → (φτ)) → φ) → φ)
6 merco1lem11 1492 . . 3 ((((χ → (φτ)) → φ) → φ) → ((((ψφ) → (((χ → (φτ)) → φ) → ⊥ )) → ⊥ ) → φ))
75, 6ax-mp 5 . 2 ((((ψφ) → (((χ → (φτ)) → φ) → ⊥ )) → ⊥ ) → φ)
8 merco1 1478 . 2 (((((ψφ) → (((χ → (φτ)) → φ) → ⊥ )) → ⊥ ) → φ) → ((φψ) → (((χ → (φτ)) → φ) → ψ)))
97, 8ax-mp 5 1 ((φψ) → (((χ → (φτ)) → φ) → ψ))
 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:  merco1lem13  1494  merco1lem14  1495
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