Theorem necon2bbid

Description: Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

	Ref	Expression
Hypothesis	necon2bbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝐴 ≠ 𝐵 ) )
Assertion	necon2bbid	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon2bbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝐴 ≠ 𝐵 ) )
2		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3	1 2	bitr3di	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓 ) )
4	3	necon4abid	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ¬ 𝜓 ) )