Theorem neor

Description: Logical OR with an equality. (Contributed by NM, 29-Apr-2007)

	Ref	Expression
Assertion	neor	⊢ ( ( 𝐴 = 𝐵 ∨ 𝜓 ) ↔ ( 𝐴 ≠ 𝐵 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		df-or	⊢ ( ( 𝐴 = 𝐵 ∨ 𝜓 ) ↔ ( ¬ 𝐴 = 𝐵 → 𝜓 ) )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	2	imbi1i	⊢ ( ( 𝐴 ≠ 𝐵 → 𝜓 ) ↔ ( ¬ 𝐴 = 𝐵 → 𝜓 ) )
4	1 3	bitr4i	⊢ ( ( 𝐴 = 𝐵 ∨ 𝜓 ) ↔ ( 𝐴 ≠ 𝐵 → 𝜓 ) )