Theorem necon2bd

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007)

	Ref	Expression
Hypothesis	necon2bd.1	$$⊢\phi \to \left(\psi \to A\ne B\right)$$
Assertion	necon2bd	$$⊢\phi \to \left(A=B\to \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		necon2bd.1	$$⊢\phi \to \left(\psi \to A\ne B\right)$$
2		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
3	1 2	syl6ib	$$⊢\phi \to \left(\psi \to \neg A=B\right)$$
4	3	con2d	$$⊢\phi \to \left(A=B\to \neg \psi \right)$$