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Metamath Proof Explorer

Theorem necon1ad

Description: Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Wolf Lammen, 23-Nov-2019)

		Ref	Expression
	Hypothesis	necon1ad.1	$⊢ φ \to (\neg ψ \to A = B)$
	Assertion	necon1ad	$⊢ φ \to (A \neq B \to ψ)$

Step	Hyp	Ref	Expression
1		necon1ad.1	$⊢ φ \to (\neg ψ \to A = B)$
2	1	necon3ad	$⊢ φ \to (A \neq B \to \neg \neg ψ)$
3		notnotr	$⊢ \neg \neg ψ \to ψ$
4	2 3	syl6	$⊢ φ \to (A \neq B \to ψ)$