Theorem re2luk2 1682

Description: luk-2 1572 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re2luk2	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Proof of Theorem re2luk2

Step	Hyp	Ref	Expression
1		rb-ax4 1671	. . . 4 ⊢ (¬ (𝜑 ∨ 𝜑) ∨ 𝜑)
2		rb-ax3 1670	. . . . . . 7 ⊢ (¬ 𝜑 ∨ (𝜑 ∨ 𝜑))
3	1, 2	rbsyl 1672	. . . . . 6 ⊢ (¬ 𝜑 ∨ 𝜑)
4		rb-ax4 1671	. . . . . . . . 9 ⊢ (¬ (¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ ¬ ¬ 𝜑)
5		rb-ax3 1670	. . . . . . . . 9 ⊢ (¬ ¬ ¬ 𝜑 ∨ (¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑))
6	4, 5	rbsyl 1672	. . . . . . . 8 ⊢ (¬ ¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑)
7		rb-ax2 1669	. . . . . . . 8 ⊢ (¬ (¬ ¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑))
8	6, 7	anmp 1667	. . . . . . 7 ⊢ (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑)
9	8, 3	rblem1 1673	. . . . . 6 ⊢ (¬ (¬ 𝜑 ∨ 𝜑) ∨ (¬ ¬ ¬ 𝜑 ∨ 𝜑))
10	3, 9	anmp 1667	. . . . 5 ⊢ (¬ ¬ ¬ 𝜑 ∨ 𝜑)
11	10, 3	rblem1 1673	. . . 4 ⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜑) ∨ (𝜑 ∨ 𝜑))
12	1, 11	rbsyl 1672	. . 3 ⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜑) ∨ 𝜑)
13		rb-imdf 1666	. . . 4 ⊢ ¬ (¬ (¬ (¬ 𝜑 → 𝜑) ∨ (¬ ¬ 𝜑 ∨ 𝜑)) ∨ ¬ (¬ (¬ ¬ 𝜑 ∨ 𝜑) ∨ (¬ 𝜑 → 𝜑)))
14	13	rblem6 1678	. . 3 ⊢ (¬ (¬ 𝜑 → 𝜑) ∨ (¬ ¬ 𝜑 ∨ 𝜑))
15	12, 14	rbsyl 1672	. 2 ⊢ (¬ (¬ 𝜑 → 𝜑) ∨ 𝜑)
16		rb-imdf 1666	. . 3 ⊢ ¬ (¬ (¬ ((¬ 𝜑 → 𝜑) → 𝜑) ∨ (¬ (¬ 𝜑 → 𝜑) ∨ 𝜑)) ∨ ¬ (¬ (¬ (¬ 𝜑 → 𝜑) ∨ 𝜑) ∨ ((¬ 𝜑 → 𝜑) → 𝜑)))
17	16	rblem7 1679	. 2 ⊢ (¬ (¬ (¬ 𝜑 → 𝜑) ∨ 𝜑) ∨ ((¬ 𝜑 → 𝜑) → 𝜑))
18	15, 17	anmp 1667	1 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382

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 196  df-or 384  df-an 385

This theorem is referenced by: (None)