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Metamath Proof Explorer

Theorem necon2abii

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007)

		Ref	Expression
	Hypothesis	necon2abii.1	$⊢ A = B \leftrightarrow \neg φ$
	Assertion	necon2abii	$⊢ φ \leftrightarrow A \neq B$

Step	Hyp	Ref	Expression
1		necon2abii.1	$⊢ A = B \leftrightarrow \neg φ$
2	1	bicomi	$⊢ \neg φ \leftrightarrow A = B$
3	2	necon1abii	$⊢ A \neq B \leftrightarrow φ$
4	3	bicomi	$⊢ φ \leftrightarrow A \neq B$