Metamath Proof Explorer


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 )