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Metamath Proof Explorer

Theorem necon2bd

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007)

		Ref	Expression
	Hypothesis	necon2bd.1	$⊢ φ \to (ψ \to A \neq B)$
	Assertion	necon2bd	$⊢ φ \to (A = B \to \neg ψ)$

Step	Hyp	Ref	Expression
1		necon2bd.1	$⊢ φ \to (ψ \to A \neq B)$
2		df-ne	$⊢ A \neq B \leftrightarrow \neg A = B$
3	1 2	imbitrdi	$⊢ φ \to (ψ \to \neg A = B)$
4	3	con2d	$⊢ φ \to (A = B \to \neg ψ)$