Theorem pm2.61ne

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 25-Nov-2019)

	Ref	Expression
Hypotheses	pm2.61ne.1	$$⊢A=B\to \left(\psi \leftrightarrow \chi \right)$$
	pm2.61ne.2	$$⊢\left(\phi \wedge A\ne B\right)\to \psi$$
	pm2.61ne.3	$$⊢\phi \to \chi$$
Assertion	pm2.61ne	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		pm2.61ne.1	$$⊢A=B\to \left(\psi \leftrightarrow \chi \right)$$
2		pm2.61ne.2	$$⊢\left(\phi \wedge A\ne B\right)\to \psi$$
3		pm2.61ne.3	$$⊢\phi \to \chi$$
4	3 1	imbitrrid	$$⊢A=B\to \left(\phi \to \psi \right)$$
5	2	expcom	$$⊢A\ne B\to \left(\phi \to \psi \right)$$
6	4 5	pm2.61ine	$$⊢\phi \to \psi$$