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Metamath Proof Explorer

Theorem necon3bi

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

		Ref	Expression
	Hypothesis	necon3bi.1	$⊢ A = B \to φ$
	Assertion	necon3bi	$⊢ \neg φ \to A \neq B$

Proof

Step	Hyp	Ref	Expression
1		necon3bi.1	$⊢ A = B \to φ$
2	1	con3i	$⊢ \neg φ \to \neg A = B$
3	2	neqned	$⊢ \neg φ \to A \neq B$