Theorem necon1abii

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007) (Proof shortened by Wolf Lammen, 25-Nov-2019)

	Ref	Expression
Hypothesis	necon1abii.1	⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )
Assertion	necon1abii	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		necon1abii.1	⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )
2		notnotb	⊢ ( 𝜑 ↔ ¬ ¬ 𝜑 )
3	1	necon3bbii	⊢ ( ¬ ¬ 𝜑 ↔ 𝐴 ≠ 𝐵 )
4	2 3	bitr2i	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )