Theorem necon3abid

Description: Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007)

	Ref	Expression
Hypothesis	necon3abid.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝜓 ) )
Assertion	necon3abid	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon3abid.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝜓 ) )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	1	notbid	⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 ↔ ¬ 𝜓 ) )
4	2 3	syl5bb	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜓 ) )