Theorem necon3i

Description: Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006) (Proof shortened by Wolf Lammen, 22-Nov-2019)

	Ref	Expression
Hypothesis	necon3i.1	⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐷 )
Assertion	necon3i	⊢ ( 𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon3i.1	⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐷 )
2	1	necon3ai	⊢ ( 𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵 )
3	2	neqned	⊢ ( 𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵 )