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Theorem ifnmfalse 28364
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 vs. applying iffalse 3738 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
ifnmfalse  |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )

Proof of Theorem ifnmfalse
StepHypRef Expression
1 df-nel 2601 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 iffalse 3738 . 2  |-  ( -.  A  e.  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
31, 2sylbi 188 1  |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    e/ wnel 2599   ifcif 3731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nel 2601  df-if 3732
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