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

Metamath Proof Explorer

Theorem ifnefalse

Description: When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse directly in this case. It happens, e.g., in oevn0 . (Contributed by David A. Wheeler, 15-May-2015)

		Ref	Expression
	Assertion	ifnefalse	$⊢ A \neq B \to if (A = B, C, D) = D$

Proof

Step	Hyp	Ref	Expression
1		df-ne	$⊢ A \neq B \leftrightarrow \neg A = B$
2		iffalse	$⊢ \neg A = B \to if (A = B, C, D) = D$
3	1 2	sylbi	$⊢ A \neq B \to if (A = B, C, D) = D$