Theorem mercolem7 1508

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem7	⊢ ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → ψ)))

Proof of Theorem mercolem7

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 ⊢ (((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ))))
2		mercolem3 1504	. . . 4 ⊢ (((φ → χ) → (θ → ψ)) → ((φ → χ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))))
3		mercolem6 1507	. . . 4 ⊢ ((((φ → χ) → (θ → ψ)) → ((φ → χ) → (((φ → χ) → (θ → ψ)) → (θ → ψ)))) → ((φ → χ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))))
4	2, 3	ax-mp 5	. . 3 ⊢ ((φ → χ) → (((φ → χ) → (θ → ψ)) → (θ → ψ)))
5		mercolem5 1506	. . . 4 ⊢ (φ → ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))))
6		mercolem4 1505	. . . 4 ⊢ ((φ → ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → ψ)))) → (((φ → χ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))))))
7	5, 6	ax-mp 5	. . 3 ⊢ (((φ → χ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → ψ)))))
8	4, 7	ax-mp 5	. 2 ⊢ ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))))
9	1, 8	ax-mp 5	1 ⊢ ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → 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:  mercolem8  1509