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	$$⊢A=B\to \neg \phi$$
Assertion	necon2ai	$$⊢\phi \to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		necon2ai.1	$$⊢A=B\to \neg \phi$$
2	1	con2i	$$⊢\phi \to \neg A=B$$
3	2	neqned	$$⊢\phi \to A\ne B$$