Theorem nemtbir

Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007)

	Ref	Expression
Hypotheses	nemtbir.1	$$⊢A\ne B$$
	nemtbir.2	$$⊢\phi \leftrightarrow A=B$$
Assertion	nemtbir	$$⊢\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		nemtbir.1	$$⊢A\ne B$$
2		nemtbir.2	$$⊢\phi \leftrightarrow A=B$$
3	1	neii	$$⊢\neg A=B$$
4	3 2	mtbir	$$⊢\neg \phi$$