Theorem re1tbw2 1749

Description: tbw-ax2 1704 rederived from merco2 1739. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1tbw2	⊢ (𝜑 → (𝜓 → 𝜑))

Proof of Theorem re1tbw2

Step	Hyp	Ref	Expression
1		mercolem1 1740	. . . 4 ⊢ (((𝜑 → 𝜑) → 𝜑) → (𝜑 → (𝜓 → 𝜑)))
2		mercolem1 1740	. . . 4 ⊢ ((((𝜑 → 𝜑) → 𝜑) → (𝜑 → (𝜓 → 𝜑))) → (𝜑 → (𝜓 → (𝜑 → (𝜓 → 𝜑)))))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 → (𝜓 → 𝜑))))
4		mercolem6 1745	. . 3 ⊢ ((𝜑 → (𝜓 → (𝜑 → (𝜓 → 𝜑)))) → (𝜓 → (𝜑 → (𝜓 → 𝜑))))
5	3, 4	ax-mp 5	. 2 ⊢ (𝜓 → (𝜑 → (𝜓 → 𝜑)))
6		mercolem6 1745	. 2 ⊢ ((𝜓 → (𝜑 → (𝜓 → 𝜑))) → (𝜑 → (𝜓 → 𝜑)))
7	5, 6	ax-mp 5	1 ⊢ (𝜑 → (𝜓 → 𝜑))

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 1542  df-fal 1552

This theorem is referenced by:  re1tbw4  1751  ltrneq  38163