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	$$⊢A\notin B\to if\left(A\in B,C,D\right)=D$$

Proof

Step	Hyp	Ref	Expression
1		df-nel	$$⊢A\notin B\leftrightarrow \neg A\in B$$
2		iffalse	$$⊢\neg A\in B\to if\left(A\in B,C,D\right)=D$$
3	1 2	sylbi	$$⊢A\notin B\to if\left(A\in B,C,D\right)=D$$