Theorem necon3abii

Description: Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007)

	Ref	Expression
Hypothesis	necon3abii.1	⊢ ( 𝐴 = 𝐵 ↔ 𝜑 )
Assertion	necon3abii	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		necon3abii.1	⊢ ( 𝐴 = 𝐵 ↔ 𝜑 )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	2 1	xchbinx	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜑 )