Theorem merco1lem3 1483

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
merco1lem3	⊢ (((φ → ψ) → (χ → ⊥ )) → (χ → φ))

Proof of Theorem merco1lem3

Step	Hyp	Ref	Expression
1		merco1lem2 1482	. . 3 ⊢ (((φ → φ) → ⊥ ) → (((φ → φ) → (φ → ⊥ )) → ⊥ ))
2		retbwax2 1481	. . . 4 ⊢ ((((φ → φ) → (φ → ⊥ )) → (φ → φ)) → (φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ))))
3		merco1lem2 1482	. . . 4 ⊢ (((((φ → φ) → (φ → ⊥ )) → (φ → φ)) → (φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ)))) → ((((φ → φ) → ⊥ ) → (((φ → φ) → (φ → ⊥ )) → ⊥ )) → (φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ)))))
4	2, 3	ax-mp 5	. . 3 ⊢ ((((φ → φ) → ⊥ ) → (((φ → φ) → (φ → ⊥ )) → ⊥ )) → (φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ))))
5	1, 4	ax-mp 5	. 2 ⊢ (φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ)))
6		merco1lem2 1482	. . 3 ⊢ (((χ → φ) → ⊥ ) → (((φ → ψ) → (χ → ⊥ )) → ⊥ ))
7		retbwax2 1481	. . . 4 ⊢ ((((φ → ψ) → (χ → ⊥ )) → (χ → φ)) → ((φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ))) → (((φ → ψ) → (χ → ⊥ )) → (χ → φ))))
8		merco1lem2 1482	. . . 4 ⊢ (((((φ → ψ) → (χ → ⊥ )) → (χ → φ)) → ((φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ))) → (((φ → ψ) → (χ → ⊥ )) → (χ → φ)))) → ((((χ → φ) → ⊥ ) → (((φ → ψ) → (χ → ⊥ )) → ⊥ )) → ((φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ))) → (((φ → ψ) → (χ → ⊥ )) → (χ → φ)))))
9	7, 8	ax-mp 5	. . 3 ⊢ ((((χ → φ) → ⊥ ) → (((φ → ψ) → (χ → ⊥ )) → ⊥ )) → ((φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ))) → (((φ → ψ) → (χ → ⊥ )) → (χ → φ))))
10	6, 9	ax-mp 5	. 2 ⊢ ((φ → (((φ → φ) → (φ → ⊥ )) → (φ → φ))) → (((φ → ψ) → (χ → ⊥ )) → (χ → φ)))
11	5, 10	ax-mp 5	1 ⊢ (((φ → ψ) → (χ → ⊥ )) → (χ → 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:  merco1lem4  1484  merco1lem6  1486  merco1lem11  1492  merco1lem12  1493  merco1lem18  1499