Theorem necon1bbid

Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008)

	Ref	Expression
Hypothesis	necon1bbid.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ 𝜓 ) )
Assertion	necon1bbid	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon1bbid.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ 𝜓 ) )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	2 1	bitr3id	⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 ↔ 𝜓 ) )
4	3	con1bid	⊢ ( 𝜑 → ( ¬ 𝜓 ↔ 𝐴 = 𝐵 ) )