Theorem necon2ad

Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypothesis	necon2ad.1	$$⊢\phi \to \left(A=B\to \neg \psi \right)$$
Assertion	necon2ad	$$⊢\phi \to \left(\psi \to A\ne B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon2ad.1	$$⊢\phi \to \left(A=B\to \neg \psi \right)$$
2		notnot	$$⊢\psi \to \neg \neg \psi$$
3	1	necon3bd	$$⊢\phi \to \left(\neg \neg \psi \to A\ne B\right)$$
4	2 3	syl5	$$⊢\phi \to \left(\psi \to A\ne B\right)$$