Theorem necon1abid

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

	Ref	Expression
Hypothesis	necon1abid.1	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ 𝐴 = 𝐵 ) )
Assertion	necon1abid	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon1abid.1	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ 𝐴 = 𝐵 ) )
2		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3	1	necon3bbid	⊢ ( 𝜑 → ( ¬ ¬ 𝜓 ↔ 𝐴 ≠ 𝐵 ) )
4	2 3	bitr2id	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ 𝜓 ) )