Theorem necon4bd

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypothesis	necon4bd.1	$$⊢\phi \to \left(\neg \psi \to A\ne B\right)$$
Assertion	necon4bd	$$⊢\phi \to \left(A=B\to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		necon4bd.1	$$⊢\phi \to \left(\neg \psi \to A\ne B\right)$$
2	1	necon2bd	$$⊢\phi \to \left(A=B\to \neg \neg \psi \right)$$
3		notnotr	$$⊢\neg \neg \psi \to \psi$$
4	2 3	syl6	$$⊢\phi \to \left(A=B\to \psi \right)$$