Theorem necon2ai

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

	Ref	Expression
Hypothesis	necon2ai.1	⊢ ( 𝐴 = 𝐵 → ¬ 𝜑 )
Assertion	necon2ai	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		necon2ai.1	⊢ ( 𝐴 = 𝐵 → ¬ 𝜑 )
2	1	con2i	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )
3	2	neqned	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )