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Theorem merco1lem13 1494
 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
merco1lem13 ((((φψ) → (χψ)) → τ) → (φτ))

Proof of Theorem merco1lem13
StepHypRef Expression
1 merco1 1478 . . . 4 (((((ψφ) → (χ → ⊥ )) → φ) → φ) → ((φψ) → (χψ)))
2 merco1lem4 1484 . . . 4 ((((((ψφ) → (χ → ⊥ )) → φ) → φ) → ((φψ) → (χψ))) → (φ → ((φψ) → (χψ))))
31, 2ax-mp 5 . . 3 (φ → ((φψ) → (χψ)))
4 merco1lem12 1493 . . 3 ((φ → ((φψ) → (χψ))) → ((((τφ) → (φ → ⊥ )) → φ) → ((φψ) → (χψ))))
53, 4ax-mp 5 . 2 ((((τφ) → (φ → ⊥ )) → φ) → ((φψ) → (χψ)))
6 merco1 1478 . 2 (((((τφ) → (φ → ⊥ )) → φ) → ((φψ) → (χψ))) → ((((φψ) → (χψ)) → τ) → (φτ)))
75, 6ax-mp 5 1 ((((φψ) → (χψ)) → τ) → (φτ))
 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:  merco1lem14  1495  merco1lem15  1496  retbwax1  1500
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