Theorem pm2.61ine

Description: Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	pm2.61ine.1	$$⊢A=B\to \phi$$
	pm2.61ine.2	$$⊢A\ne B\to \phi$$
Assertion	pm2.61ine	$$⊢\phi$$

Proof

Step	Hyp	Ref	Expression
1		pm2.61ine.1	$$⊢A=B\to \phi$$
2		pm2.61ine.2	$$⊢A\ne B\to \phi$$
3		nne	$$⊢\neg A\ne B\leftrightarrow A=B$$
4	3 1	sylbi	$$⊢\neg A\ne B\to \phi$$
5	2 4	pm2.61i	$$⊢\phi$$