Theorem re1tbw1 1751

Description: tbw-ax1 1706 rederived from merco2 1742. (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 1750	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))))
2		mercolem3 1745	. . 3 ⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))
3		mercolem6 1748	. . 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 68	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))))
5		mercolem6 1748	. 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 206  df-tru 1544  df-fal 1554

This theorem is referenced by:  re1tbw4  1754