Theorem necon3bi

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

	Ref	Expression
Hypothesis	necon3bi.1	$$⊢A=B\to \phi$$
Assertion	necon3bi	$$⊢\neg \phi \to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		necon3bi.1	$$⊢A=B\to \phi$$
2	1	con3i	$$⊢\neg \phi \to \neg A=B$$
3	2	neqned	$$⊢\neg \phi \to A\ne B$$