Theorem rb-ax4

Description: The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	rb-ax4	⊢ ( ¬ ( 𝜑 ∨ 𝜑 ) ∨ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		pm1.2	⊢ ( ( 𝜑 ∨ 𝜑 ) → 𝜑 )
2	1	con3i	⊢ ( ¬ 𝜑 → ¬ ( 𝜑 ∨ 𝜑 ) )
3	2	con1i	⊢ ( ¬ ¬ ( 𝜑 ∨ 𝜑 ) → 𝜑 )
4	3	orri	⊢ ( ¬ ( 𝜑 ∨ 𝜑 ) ∨ 𝜑 )