Theorem pm2.61dne

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	pm2.61dne.1	$$⊢\phi \to \left(A=B\to \psi \right)$$
	pm2.61dne.2	$$⊢\phi \to \left(A\ne B\to \psi \right)$$
Assertion	pm2.61dne	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		pm2.61dne.1	$$⊢\phi \to \left(A=B\to \psi \right)$$
2		pm2.61dne.2	$$⊢\phi \to \left(A\ne B\to \psi \right)$$
3		nne	$$⊢\neg A\ne B\leftrightarrow A=B$$
4	3 1	syl5bi	$$⊢\phi \to \left(\neg A\ne B\to \psi \right)$$
5	2 4	pm2.61d	$$⊢\phi \to \psi$$