Theorem necon2d

Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008)

	Ref	Expression
Hypothesis	necon2d.1	$$⊢\phi \to \left(A=B\to C\ne D\right)$$
Assertion	necon2d	$$⊢\phi \to \left(C=D\to A\ne B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon2d.1	$$⊢\phi \to \left(A=B\to C\ne D\right)$$
2		df-ne	$$⊢C\ne D\leftrightarrow \neg C=D$$
3	1 2	imbitrdi	$$⊢\phi \to \left(A=B\to \neg C=D\right)$$
4	3	necon2ad	$$⊢\phi \to \left(C=D\to A\ne B\right)$$