Theorem rblem6

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.)

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

Proof

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