Theorem necon2abii

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007)

	Ref	Expression
Hypothesis	necon2abii.1	⊢ ( 𝐴 = 𝐵 ↔ ¬ 𝜑 )
Assertion	necon2abii	⊢ ( 𝜑 ↔ 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon2abii.1	⊢ ( 𝐴 = 𝐵 ↔ ¬ 𝜑 )
2	1	bicomi	⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )
3	2	necon1abii	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )
4	3	bicomi	⊢ ( 𝜑 ↔ 𝐴 ≠ 𝐵 )