Theorem necon4bid

Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007)

	Ref	Expression
Hypothesis	necon4bid.1	$$⊢\phi \to \left(A\ne B\leftrightarrow C\ne D\right)$$
Assertion	necon4bid	$$⊢\phi \to \left(A=B\leftrightarrow C=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		necon4bid.1	$$⊢\phi \to \left(A\ne B\leftrightarrow C\ne D\right)$$
2	1	necon2bbid	$$⊢\phi \to \left(C=D\leftrightarrow \neg A\ne B\right)$$
3		nne	$$⊢\neg A\ne B\leftrightarrow A=B$$
4	2 3	bitr2di	$$⊢\phi \to \left(A=B\leftrightarrow C=D\right)$$