Theorem necon3d

Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006)

	Ref	Expression
Hypothesis	necon3d.1	$$⊢\phi \to \left(A=B\to C=D\right)$$
Assertion	necon3d	$$⊢\phi \to \left(C\ne D\to A\ne B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon3d.1	$$⊢\phi \to \left(A=B\to C=D\right)$$
2	1	necon3ad	$$⊢\phi \to \left(C\ne D\to \neg A=B\right)$$
3		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
4	2 3	syl6ibr	$$⊢\phi \to \left(C\ne D\to A\ne B\right)$$