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

Proof of Theorem merco1lem12

Step	Hyp	Ref	Expression
1		merco1lem3 1483	. . . . 5 ⊢ ((((φ → τ) → (((χ → (φ → τ)) → φ) → ⊥ )) → (χ → ⊥ )) → (χ → (φ → τ)))
2		merco1 1478	. . . . 5 ⊢ (((((φ → τ) → (((χ → (φ → τ)) → φ) → ⊥ )) → (χ → ⊥ )) → (χ → (φ → τ))) → (((χ → (φ → τ)) → φ) → (((χ → (φ → τ)) → φ) → φ)))
3	1, 2	ax-mp 5	. . . 4 ⊢ (((χ → (φ → τ)) → φ) → (((χ → (φ → τ)) → φ) → φ))
4		merco1lem9 1490	. . . 4 ⊢ ((((χ → (φ → τ)) → φ) → (((χ → (φ → τ)) → φ) → φ)) → (((χ → (φ → τ)) → φ) → φ))
5	3, 4	ax-mp 5	. . 3 ⊢ (((χ → (φ → τ)) → φ) → φ)
6		merco1lem11 1492	. . 3 ⊢ ((((χ → (φ → τ)) → φ) → φ) → ((((ψ → φ) → (((χ → (φ → τ)) → φ) → ⊥ )) → ⊥ ) → φ))
7	5, 6	ax-mp 5	. 2 ⊢ ((((ψ → φ) → (((χ → (φ → τ)) → φ) → ⊥ )) → ⊥ ) → φ)
8		merco1 1478	. 2 ⊢ (((((ψ → φ) → (((χ → (φ → τ)) → φ) → ⊥ )) → ⊥ ) → φ) → ((φ → ψ) → (((χ → (φ → τ)) → φ) → ψ)))
9	7, 8	ax-mp 5	1 ⊢ ((φ → ψ) → (((χ → (φ → τ)) → φ) → ψ))

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