Theorem necon4abid

Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008) (Proof shortened by Wolf Lammen, 24-Nov-2019)

	Ref	Expression
Hypothesis	necon4abid.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜓 ) )
Assertion	necon4abid	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon4abid.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜓 ) )
2		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3	1	necon1bbid	⊢ ( 𝜑 → ( ¬ ¬ 𝜓 ↔ 𝐴 = 𝐵 ) )
4	2 3	bitr2id	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝜓 ) )