Theorem necon3ai

Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypothesis	necon3ai.1	$$⊢\phi \to A=B$$
Assertion	necon3ai	$$⊢A\ne B\to \neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		necon3ai.1	$$⊢\phi \to A=B$$
2		nne	$$⊢\neg A\ne B\leftrightarrow A=B$$
3	1 2	sylibr	$$⊢\phi \to \neg A\ne B$$
4	3	con2i	$$⊢A\ne B\to \neg \phi$$