Theorem necon1bbii

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

	Ref	Expression
Hypothesis	necon1bbii.1	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )
Assertion	necon1bbii	⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon1bbii.1	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )
2		nne	⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
3	2 1	xchnxbi	⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )