Theorem necon1ad

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

	Ref	Expression
Hypothesis	necon1ad.1	$$⊢\phi \to \left(\neg \psi \to A=B\right)$$
Assertion	necon1ad	$$⊢\phi \to \left(A\ne B\to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		necon1ad.1	$$⊢\phi \to \left(\neg \psi \to A=B\right)$$
2	1	necon3ad	$$⊢\phi \to \left(A\ne B\to \neg \neg \psi \right)$$
3		notnotr	$$⊢\neg \neg \psi \to \psi$$
4	2 3	syl6	$$⊢\phi \to \left(A\ne B\to \psi \right)$$