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Metamath Proof Explorer

Theorem necon1bi

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

		Ref	Expression
	Hypothesis	necon1bi.1	$⊢ A \neq B \to φ$
	Assertion	necon1bi	$⊢ \neg φ \to A = B$

Proof

Step	Hyp	Ref	Expression
1		necon1bi.1	$⊢ A \neq B \to φ$
2		df-ne	$⊢ A \neq B \leftrightarrow \neg A = B$
3	2 1	sylbir	$⊢ \neg A = B \to φ$
4	3	con1i	$⊢ \neg φ \to A = B$