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Metamath Proof Explorer

Theorem necon2ai

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

		Ref	Expression
	Hypothesis	necon2ai.1	$⊢ A = B \to \neg φ$
	Assertion	necon2ai	$⊢ φ \to A \neq B$

Proof

Step	Hyp	Ref	Expression
1		necon2ai.1	$⊢ A = B \to \neg φ$
2	1	con2i	$⊢ φ \to \neg A = B$
3	2	neqned	$⊢ φ \to A \neq B$