Theorem necon2bbii

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007)

	Ref	Expression
Hypothesis	necon2bbii.1	⊢ ( 𝜑 ↔ 𝐴 ≠ 𝐵 )
Assertion	necon2bbii	⊢ ( 𝐴 = 𝐵 ↔ ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		necon2bbii.1	⊢ ( 𝜑 ↔ 𝐴 ≠ 𝐵 )
2	1	bicomi	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )
3	2	necon1bbii	⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )
4	3	bicomi	⊢ ( 𝐴 = 𝐵 ↔ ¬ 𝜑 )