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	⊢ ((((φ → ψ) → (χ → ψ)) → τ) → (φ → τ))

Proof of Theorem merco1lem13

Step	Hyp	Ref	Expression
1		merco1 1478	. . . 4 ⊢ (((((ψ → φ) → (χ → ⊥ )) → φ) → φ) → ((φ → ψ) → (χ → ψ)))
2		merco1lem4 1484	. . . 4 ⊢ ((((((ψ → φ) → (χ → ⊥ )) → φ) → φ) → ((φ → ψ) → (χ → ψ))) → (φ → ((φ → ψ) → (χ → ψ))))
3	1, 2	ax-mp 5	. . 3 ⊢ (φ → ((φ → ψ) → (χ → ψ)))
4		merco1lem12 1493	. . 3 ⊢ ((φ → ((φ → ψ) → (χ → ψ))) → ((((τ → φ) → (φ → ⊥ )) → φ) → ((φ → ψ) → (χ → ψ))))
5	3, 4	ax-mp 5	. 2 ⊢ ((((τ → φ) → (φ → ⊥ )) → φ) → ((φ → ψ) → (χ → ψ)))
6		merco1 1478	. 2 ⊢ (((((τ → φ) → (φ → ⊥ )) → φ) → ((φ → ψ) → (χ → ψ))) → ((((φ → ψ) → (χ → ψ)) → τ) → (φ → τ)))
7	5, 6	ax-mp 5	1 ⊢ ((((φ → ψ) → (χ → ψ)) → τ) → (φ → τ))

This proof depends on syntax axioms:   → wi 4   ⊥ wfal 1317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320

This theorem is used by:  merco1lem14  1495  merco1lem15  1496  retbwax1  1500