Theorem ifnmfalse

Description: If A is not a member of B, but an "if" condition requires it, 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. (Contributed by David A. Wheeler, 15-May-2015)

	Ref	Expression
Assertion	ifnmfalse	|- ( A e/ B -> if ( A e. B , C , D ) = D )

Proof

Step	Hyp	Ref	Expression
1		df-nel	|- ( A e/ B <-> -. A e. B )
2		iffalse	|- ( -. A e. B -> if ( A e. B , C , D ) = D )
3	1 2	sylbi	|- ( A e/ B -> if ( A e. B , C , D ) = D )