Theorem retbwax2 1745

Description: tbw-ax2 1730 rederived from merco1 1742. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
retbwax2	⊢ (𝜑 → (𝜓 → 𝜑))

Proof of Theorem retbwax2

Step	Hyp	Ref	Expression
1		merco1lem1 1743	. . . 4 ⊢ (((((𝜑 → 𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑))
2		merco1 1742	. . . 4 ⊢ ((((((𝜑 → 𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → (𝜑 → 𝜑)) → (𝜑 → (𝜑 → 𝜑))))
3	1, 2	ax-mp 5	. . 3 ⊢ (((⊥ → 𝜑) → (𝜑 → 𝜑)) → (𝜑 → (𝜑 → 𝜑)))
4		merco1 1742	. . . 4 ⊢ (((((𝜑 → (𝜑 → 𝜑)) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜑 → 𝜑)))
5		merco1 1742	. . . 4 ⊢ ((((((𝜑 → (𝜑 → 𝜑)) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜑 → 𝜑))) → ((((⊥ → 𝜑) → (𝜑 → 𝜑)) → (𝜑 → (𝜑 → 𝜑))) → (𝜑 → (𝜑 → (𝜑 → 𝜑)))))
6	4, 5	ax-mp 5	. . 3 ⊢ ((((⊥ → 𝜑) → (𝜑 → 𝜑)) → (𝜑 → (𝜑 → 𝜑))) → (𝜑 → (𝜑 → (𝜑 → 𝜑))))
7	3, 6	ax-mp 5	. 2 ⊢ (𝜑 → (𝜑 → (𝜑 → 𝜑)))
8		merco1lem1 1743	. . . 4 ⊢ (((((𝜓 → 𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑))
9		merco1 1742	. . . 4 ⊢ ((((((𝜓 → 𝜑) → 𝜑) → (𝜑 → ⊥)) → 𝜑) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → (𝜓 → 𝜑)) → (𝜑 → (𝜓 → 𝜑))))
10	8, 9	ax-mp 5	. . 3 ⊢ (((⊥ → 𝜑) → (𝜓 → 𝜑)) → (𝜑 → (𝜓 → 𝜑)))
11		merco1 1742	. . . 4 ⊢ (((((𝜑 → (𝜓 → 𝜑)) → (𝜓 → ⊥)) → ((𝜑 → (𝜑 → (𝜑 → 𝜑))) → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜓 → 𝜑)))
12		merco1 1742	. . . 4 ⊢ ((((((𝜑 → (𝜓 → 𝜑)) → (𝜓 → ⊥)) → ((𝜑 → (𝜑 → (𝜑 → 𝜑))) → ⊥)) → ⊥) → ((⊥ → 𝜑) → (𝜓 → 𝜑))) → ((((⊥ → 𝜑) → (𝜓 → 𝜑)) → (𝜑 → (𝜓 → 𝜑))) → ((𝜑 → (𝜑 → (𝜑 → 𝜑))) → (𝜑 → (𝜓 → 𝜑)))))
13	11, 12	ax-mp 5	. . 3 ⊢ ((((⊥ → 𝜑) → (𝜓 → 𝜑)) → (𝜑 → (𝜓 → 𝜑))) → ((𝜑 → (𝜑 → (𝜑 → 𝜑))) → (𝜑 → (𝜓 → 𝜑))))
14	10, 13	ax-mp 5	. 2 ⊢ ((𝜑 → (𝜑 → (𝜑 → 𝜑))) → (𝜑 → (𝜓 → 𝜑)))
15	7, 14	ax-mp 5	1 ⊢ (𝜑 → (𝜓 → 𝜑))

This proof depends on syntax axioms:   → wi 4  ⊥wfal 1581

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 1572  df-fal 1582

This theorem is used by:  merco1lem2  1746  merco1lem3  1747  retbwax3  1752