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Metamath Proof Explorer

Theorem necon4bid

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

		Ref	Expression
	Hypothesis	necon4bid.1	$⊢ φ \to (A \neq B \leftrightarrow C \neq D)$
	Assertion	necon4bid	$⊢ φ \to (A = B \leftrightarrow C = D)$

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