Theorem re1tbw3 1780

Description: tbw-ax3 1735 rederived from merco2 1769. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1tbw3	⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

Proof of Theorem re1tbw3

Step	Hyp	Ref	Expression
1		mercolem2 1771	. 2 ⊢ (((𝜑 → 𝜑) → 𝜑) → (𝜑 → (𝜑 → 𝜑)))
2		mercolem2 1771	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜑) → ((((𝜑 → 𝜑) → 𝜑) → (𝜑 → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)))
3		mercolem6 1775	. . 3 ⊢ ((((𝜑 → 𝜓) → 𝜑) → ((((𝜑 → 𝜑) → 𝜑) → (𝜑 → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))) → ((((𝜑 → 𝜑) → 𝜑) → (𝜑 → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → 𝜑) → 𝜑)))
4	2, 3	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜑) → 𝜑) → (𝜑 → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
5	1, 4	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-tru 1573  df-fal 1583

This theorem is used by:  re1tbw4  1781