Theorem necon3ad

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypothesis	necon3ad.1	$$⊢\phi \to \left(\psi \to A=B\right)$$
Assertion	necon3ad	$$⊢\phi \to \left(A\ne B\to \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		necon3ad.1	$$⊢\phi \to \left(\psi \to A=B\right)$$
2		neneq	$$⊢A\ne B\to \neg A=B$$
3	1 2	nsyli	$$⊢\phi \to \left(A\ne B\to \neg \psi \right)$$