Theorem merco1lem2 1482

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem2	⊢ (((φ → ψ) → χ) → (((ψ → τ) → (φ → ⊥ )) → χ))

Proof of Theorem merco1lem2

Step	Hyp	Ref	Expression
1		retbwax2 1481	. . 3 ⊢ ((((ψ → τ) → (φ → ⊥ )) → ⊥ ) → ((χ → φ) → (((ψ → τ) → (φ → ⊥ )) → ⊥ )))
2		merco1 1478	. . 3 ⊢ (((((ψ → τ) → (φ → ⊥ )) → ⊥ ) → ((χ → φ) → (((ψ → τ) → (φ → ⊥ )) → ⊥ ))) → ((((χ → φ) → (((ψ → τ) → (φ → ⊥ )) → ⊥ )) → ψ) → (φ → ψ)))
3	1, 2	ax-mp 5	. 2 ⊢ ((((χ → φ) → (((ψ → τ) → (φ → ⊥ )) → ⊥ )) → ψ) → (φ → ψ))
4		merco1 1478	. 2 ⊢ (((((χ → φ) → (((ψ → τ) → (φ → ⊥ )) → ⊥ )) → ψ) → (φ → ψ)) → (((φ → ψ) → χ) → (((ψ → τ) → (φ → ⊥ )) → χ)))
5	3, 4	ax-mp 5	1 ⊢ (((φ → ψ) → χ) → (((ψ → τ) → (φ → ⊥ )) → χ))

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:  merco1lem3  1483  merco1lem10  1491  merco1lem11  1492  merco1lem18  1499