Theorem necon4d

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypothesis	necon4d.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷 ) )
Assertion	necon4d	⊢ ( 𝜑 → ( 𝐶 = 𝐷 → 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon4d.1	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷 ) )
2	1	necon2bd	⊢ ( 𝜑 → ( 𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵 ) )
3		nne	⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
4	2 3	syl6ib	⊢ ( 𝜑 → ( 𝐶 = 𝐷 → 𝐴 = 𝐵 ) )