Theorem rb-ax1

Description: The first 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-ax1	⊢ ( ¬ ( ¬ 𝜓 ∨ 𝜒 ) ∨ ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		orim2	⊢ ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ 𝜒 ) ) )
2		imor	⊢ ( ( 𝜓 → 𝜒 ) ↔ ( ¬ 𝜓 ∨ 𝜒 ) )
3		imor	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ 𝜒 ) ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜒 ) ) )
4	1 2 3	3imtr3i	⊢ ( ( ¬ 𝜓 ∨ 𝜒 ) → ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜒 ) ) )
5	4	imori	⊢ ( ¬ ( ¬ 𝜓 ∨ 𝜒 ) ∨ ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜒 ) ) )