Theorem necon2d

Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008)

	Ref	Expression
Hypothesis	necon2d.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝐶 ≠ 𝐷 ) )
Assertion	necon2d	⊢ ( 𝜑 → ( 𝐶 = 𝐷 → 𝐴 ≠ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon2d.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝐶 ≠ 𝐷 ) )
2		df-ne	⊢ ( 𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷 )
3	1 2	imbitrdi	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ¬ 𝐶 = 𝐷 ) )
4	3	necon2ad	⊢ ( 𝜑 → ( 𝐶 = 𝐷 → 𝐴 ≠ 𝐵 ) )