Theorem merco1lem10 1491

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
merco1lem10	⊢ (((((φ → ψ) → χ) → (τ → χ)) → φ) → (θ → φ))

Proof of Theorem merco1lem10

Step	Hyp	Ref	Expression
1		merco1 1478	. . 3 ⊢ (((((χ → φ) → (τ → ⊥ )) → φ) → (φ → ψ)) → (((φ → ψ) → χ) → (τ → χ)))
2		merco1lem2 1482	. . 3 ⊢ ((((((χ → φ) → (τ → ⊥ )) → φ) → (φ → ψ)) → (((φ → ψ) → χ) → (τ → χ))) → ((((φ → ψ) → (θ → ⊥ )) → ((((χ → φ) → (τ → ⊥ )) → φ) → ⊥ )) → (((φ → ψ) → χ) → (τ → χ))))
3	1, 2	ax-mp 5	. 2 ⊢ ((((φ → ψ) → (θ → ⊥ )) → ((((χ → φ) → (τ → ⊥ )) → φ) → ⊥ )) → (((φ → ψ) → χ) → (τ → χ)))
4		merco1 1478	. 2 ⊢ (((((φ → ψ) → (θ → ⊥ )) → ((((χ → φ) → (τ → ⊥ )) → φ) → ⊥ )) → (((φ → ψ) → χ) → (τ → χ))) → (((((φ → ψ) → χ) → (τ → χ)) → φ) → (θ → φ)))
5	3, 4	ax-mp 5	1 ⊢ (((((φ → ψ) → χ) → (τ → χ)) → φ) → (θ → φ))

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:  retbwax1  1500