Theorem necon4bbid

Description: Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012)

	Ref	Expression
Hypothesis	necon4bbid.1	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ 𝐴 ≠ 𝐵 ) )
Assertion	necon4bbid	⊢ ( 𝜑 → ( 𝜓 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon4bbid.1	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ 𝐴 ≠ 𝐵 ) )
2	1	bicomd	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜓 ) )
3	2	necon4abid	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝜓 ) )
4	3	bicomd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝐴 = 𝐵 ) )