Theorem rblem5 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.)

Assertion

Ref	Expression
rblem5	⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (¬ ¬ 𝜓 ∨ 𝜑))

Proof of Theorem rblem5

Step	Hyp	Ref	Expression
1		rb-ax2 1757	. 2 ⊢ (¬ (𝜑 ∨ ¬ ¬ 𝜓) ∨ (¬ ¬ 𝜓 ∨ 𝜑))
2		rb-ax4 1759	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜑) ∨ 𝜑)
3		rb-ax3 1758	. . . . 5 ⊢ (¬ 𝜑 ∨ (𝜑 ∨ 𝜑))
4	2, 3	rbsyl 1760	. . . 4 ⊢ (¬ 𝜑 ∨ 𝜑)
5		rb-ax4 1759	. . . . . . 7 ⊢ (¬ (¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ ¬ ¬ 𝜑)
6		rb-ax3 1758	. . . . . . 7 ⊢ (¬ ¬ ¬ 𝜑 ∨ (¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑))
7	5, 6	rbsyl 1760	. . . . . 6 ⊢ (¬ ¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑)
8		rb-ax2 1757	. . . . . 6 ⊢ (¬ (¬ ¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑))
9	7, 8	anmp 1755	. . . . 5 ⊢ (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑)
10	9, 4	rblem1 1761	. . . 4 ⊢ (¬ (¬ 𝜑 ∨ 𝜑) ∨ (¬ ¬ ¬ 𝜑 ∨ 𝜑))
11	4, 10	anmp 1755	. . 3 ⊢ (¬ ¬ ¬ 𝜑 ∨ 𝜑)
12		rb-ax4 1759	. . . . 5 ⊢ (¬ (¬ 𝜓 ∨ ¬ 𝜓) ∨ ¬ 𝜓)
13		rb-ax3 1758	. . . . 5 ⊢ (¬ ¬ 𝜓 ∨ (¬ 𝜓 ∨ ¬ 𝜓))
14	12, 13	rbsyl 1760	. . . 4 ⊢ (¬ ¬ 𝜓 ∨ ¬ 𝜓)
15		rb-ax2 1757	. . . 4 ⊢ (¬ (¬ ¬ 𝜓 ∨ ¬ 𝜓) ∨ (¬ 𝜓 ∨ ¬ ¬ 𝜓))
16	14, 15	anmp 1755	. . 3 ⊢ (¬ 𝜓 ∨ ¬ ¬ 𝜓)
17	11, 16	rblem1 1761	. 2 ⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ∨ ¬ ¬ 𝜓))
18	1, 17	rbsyl 1760	1 ⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (¬ ¬ 𝜓 ∨ 𝜑))

Syntax hints:  ¬ wn 3   ∨ wo 843

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 396  df-or 844

This theorem is referenced by:  rblem6  1766  rblem7  1767