Theorem rblem6 1765

Description: Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
rblem6.1	⊢ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑))

Assertion

Ref	Expression
rblem6	⊢ (¬ 𝜑 ∨ 𝜓)

Proof of Theorem rblem6

Step	Hyp	Ref	Expression
1		rblem6.1	. 2 ⊢ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑))
2		rb-ax4 1758	. . . . . . 7 ⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ 𝜑 ∨ 𝜓))
3		rb-ax3 1757	. . . . . . 7 ⊢ (¬ ¬ (¬ 𝜑 ∨ 𝜓) ∨ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜑 ∨ 𝜓)))
4	2, 3	rbsyl 1759	. . . . . 6 ⊢ (¬ ¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜑 ∨ 𝜓))
5		rb-ax2 1756	. . . . . 6 ⊢ (¬ (¬ ¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜑 ∨ 𝜓)) ∨ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ ¬ (¬ 𝜑 ∨ 𝜓)))
6	4, 5	anmp 1754	. . . . 5 ⊢ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ ¬ (¬ 𝜑 ∨ 𝜓))
7		rblem3 1762	. . . . 5 ⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ ¬ (¬ 𝜑 ∨ 𝜓)) ∨ ((¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)) ∨ ¬ ¬ (¬ 𝜑 ∨ 𝜓)))
8	6, 7	anmp 1754	. . . 4 ⊢ ((¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)) ∨ ¬ ¬ (¬ 𝜑 ∨ 𝜓))
9		rb-ax2 1756	. . . 4 ⊢ (¬ ((¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)) ∨ ¬ ¬ (¬ 𝜑 ∨ 𝜓)) ∨ (¬ ¬ (¬ 𝜑 ∨ 𝜓) ∨ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑))))
10	8, 9	anmp 1754	. . 3 ⊢ (¬ ¬ (¬ 𝜑 ∨ 𝜓) ∨ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))
11		rblem5 1764	. . 3 ⊢ (¬ (¬ ¬ (¬ 𝜑 ∨ 𝜓) ∨ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑))) ∨ (¬ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)) ∨ (¬ 𝜑 ∨ 𝜓)))
12	10, 11	anmp 1754	. 2 ⊢ (¬ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)) ∨ (¬ 𝜑 ∨ 𝜓))
13	1, 12	anmp 1754	1 ⊢ (¬ 𝜑 ∨ 𝜓)

Syntax hints:  ¬ wn 3   ∨ wo 844

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-an 397  df-or 845

This theorem is referenced by:  re1axmp  1767  re2luk1  1768  re2luk2  1769