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Metamath Proof Explorer

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	$⊢ A = B \to φ$
	pm2.61ine.2	$⊢ A \neq B \to φ$
Assertion	pm2.61ine	$⊢ φ$

Proof

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