Theorem necon4d

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypothesis	necon4d.1	$$⊢\phi \to \left(A\ne B\to C\ne D\right)$$
Assertion	necon4d	$$⊢\phi \to \left(C=D\to A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon4d.1	$$⊢\phi \to \left(A\ne B\to C\ne D\right)$$
2	1	necon2bd	$$⊢\phi \to \left(C=D\to \neg A\ne B\right)$$
3		nne	$$⊢\neg A\ne B\leftrightarrow A=B$$
4	2 3	syl6ib	$$⊢\phi \to \left(C=D\to A=B\right)$$