Theorem rblem5 1526

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 1518	. 2 ⊢ (¬ (φ ∨ ¬ ¬ ψ) ∨ (¬ ¬ ψ ∨ φ))
2		rb-ax4 1520	. . . . 5 ⊢ (¬ (φ ∨ φ) ∨ φ)
3		rb-ax3 1519	. . . . 5 ⊢ (¬ φ ∨ (φ ∨ φ))
4	2, 3	rbsyl 1521	. . . 4 ⊢ (¬ φ ∨ φ)
5		rb-ax4 1520	. . . . . . 7 ⊢ (¬ (¬ ¬ φ ∨ ¬ ¬ φ) ∨ ¬ ¬ φ)
6		rb-ax3 1519	. . . . . . 7 ⊢ (¬ ¬ ¬ φ ∨ (¬ ¬ φ ∨ ¬ ¬ φ))
7	5, 6	rbsyl 1521	. . . . . 6 ⊢ (¬ ¬ ¬ φ ∨ ¬ ¬ φ)
8		rb-ax2 1518	. . . . . 6 ⊢ (¬ (¬ ¬ ¬ φ ∨ ¬ ¬ φ) ∨ (¬ ¬ φ ∨ ¬ ¬ ¬ φ))
9	7, 8	anmp 1516	. . . . 5 ⊢ (¬ ¬ φ ∨ ¬ ¬ ¬ φ)
10	9, 4	rblem1 1522	. . . 4 ⊢ (¬ (¬ φ ∨ φ) ∨ (¬ ¬ ¬ φ ∨ φ))
11	4, 10	anmp 1516	. . 3 ⊢ (¬ ¬ ¬ φ ∨ φ)
12		rb-ax4 1520	. . . . 5 ⊢ (¬ (¬ ψ ∨ ¬ ψ) ∨ ¬ ψ)
13		rb-ax3 1519	. . . . 5 ⊢ (¬ ¬ ψ ∨ (¬ ψ ∨ ¬ ψ))
14	12, 13	rbsyl 1521	. . . 4 ⊢ (¬ ¬ ψ ∨ ¬ ψ)
15		rb-ax2 1518	. . . 4 ⊢ (¬ (¬ ¬ ψ ∨ ¬ ψ) ∨ (¬ ψ ∨ ¬ ¬ ψ))
16	14, 15	anmp 1516	. . 3 ⊢ (¬ ψ ∨ ¬ ¬ ψ)
17	11, 16	rblem1 1522	. 2 ⊢ (¬ (¬ ¬ φ ∨ ψ) ∨ (φ ∨ ¬ ¬ ψ))
18	1, 17	rbsyl 1521	1 ⊢ (¬ (¬ ¬ φ ∨ ψ) ∨ (¬ ¬ ψ ∨ φ))

Syntax hints:  ¬ wn 3   ∨ wo 357

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 177  df-or 359  df-an 360

This theorem is referenced by:  rblem6  1527  rblem7  1528