Theorem necon1i

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007)

	Ref	Expression
Hypothesis	necon1i.1	⊢ ( 𝐴 ≠ 𝐵 → 𝐶 = 𝐷 )
Assertion	necon1i	⊢ ( 𝐶 ≠ 𝐷 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon1i.1	⊢ ( 𝐴 ≠ 𝐵 → 𝐶 = 𝐷 )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	2 1	sylbir	⊢ ( ¬ 𝐴 = 𝐵 → 𝐶 = 𝐷 )
4	3	necon1ai	⊢ ( 𝐶 ≠ 𝐷 → 𝐴 = 𝐵 )