Theorem necon4bid

Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007)

	Ref	Expression
Hypothesis	necon4bid.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) )
Assertion	necon4bid	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		necon4bid.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) )
2	1	necon2bbid	⊢ ( 𝜑 → ( 𝐶 = 𝐷 ↔ ¬ 𝐴 ≠ 𝐵 ) )
3		nne	⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
4	2 3	syl6rbb	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝐶 = 𝐷 ) )