Theorem re1tbw1 1510

Description: tbw-ax1 1465 rederived from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1tbw1	⊢ ((φ → ψ) → ((ψ → χ) → (φ → χ)))

Proof of Theorem re1tbw1

Step	Hyp	Ref	Expression
1		mercolem8 1509	. . 3 ⊢ ((φ → ψ) → ((ψ → (φ → χ)) → ((φ → ψ) → ((ψ → χ) → (φ → χ)))))
2		mercolem3 1504	. . 3 ⊢ ((ψ → χ) → (ψ → (φ → χ)))
3		mercolem6 1507	. . 3 ⊢ (((φ → ψ) → ((ψ → (φ → χ)) → ((φ → ψ) → ((ψ → χ) → (φ → χ))))) → ((ψ → (φ → χ)) → ((φ → ψ) → ((ψ → χ) → (φ → χ)))))
4	1, 2, 3	mpsyl 59	. 2 ⊢ ((ψ → χ) → ((φ → ψ) → ((ψ → χ) → (φ → χ))))
5		mercolem6 1507	. 2 ⊢ (((ψ → χ) → ((φ → ψ) → ((ψ → χ) → (φ → χ)))) → ((φ → ψ) → ((ψ → χ) → (φ → χ))))
6	4, 5	ax-mp 5	1 ⊢ ((φ → ψ) → ((ψ → χ) → (φ → χ)))

Syntax hints:   → wi 4

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:  re1tbw4  1513