Theorem merco1lem11 1492

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
merco1lem11	⊢ ((φ → ψ) → (((χ → (φ → τ)) → ⊥ ) → ψ))

Proof of Theorem merco1lem11

Step	Hyp	Ref	Expression
1		merco1lem5 1485	. . . . . 6 ⊢ ((((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → (((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ))
2		merco1lem3 1483	. . . . . 6 ⊢ (((((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → (((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ )) → (((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
3	1, 2	ax-mp 5	. . . . 5 ⊢ (((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
4		merco1lem4 1484	. . . . 5 ⊢ ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((((χ → (φ → τ)) → ⊥ ) → ⊥ ) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
5	3, 4	ax-mp 5	. . . 4 ⊢ ((((χ → (φ → τ)) → ⊥ ) → ⊥ ) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
6		merco1lem5 1485	. . . 4 ⊢ (((((χ → (φ → τ)) → ⊥ ) → ⊥ ) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((χ → (φ → τ)) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
7	5, 6	ax-mp 5	. . 3 ⊢ ((χ → (φ → τ)) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
8		merco1lem4 1484	. . 3 ⊢ (((χ → (φ → τ)) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((φ → τ) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
9	7, 8	ax-mp 5	. 2 ⊢ ((φ → τ) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
10		merco1 1478	. . 3 ⊢ (((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → φ) → ((φ → ψ) → (((χ → (φ → τ)) → ⊥ ) → ψ)))
11		merco1lem2 1482	. . 3 ⊢ ((((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → φ) → ((φ → ψ) → (((χ → (φ → τ)) → ⊥ ) → ψ))) → (((φ → τ) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((φ → ψ) → (((χ → (φ → τ)) → ⊥ ) → ψ))))
12	10, 11	ax-mp 5	. 2 ⊢ (((φ → τ) → ((((ψ → φ) → (((χ → (φ → τ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((φ → ψ) → (((χ → (φ → τ)) → ⊥ ) → ψ)))
13	9, 12	ax-mp 5	1 ⊢ ((φ → ψ) → (((χ → (φ → τ)) → ⊥ ) → ψ))

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:  merco1lem12  1493  merco1lem16  1497  merco1lem17  1498