This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem necon3ad

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 23-Nov-2019)

		Ref	Expression
	Hypothesis	necon3ad.1	$⊢ φ \to (ψ \to A = B)$
	Assertion	necon3ad	$⊢ φ \to (A \neq B \to \neg ψ)$

Proof

Step	Hyp	Ref	Expression
1		necon3ad.1	$⊢ φ \to (ψ \to A = B)$
2		neneq	$⊢ A \neq B \to \neg A = B$
3	1 2	nsyli	$⊢ φ \to (A \neq B \to \neg ψ)$