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Theorem ifnmfalse 28604
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 3770 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 2608 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 iffalse 3770 . 2  |-  ( -.  A  e.  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
31, 2sylbi 189 1  |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1727    e/ wnel 2606   ifcif 3763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nel 2608  df-if 3764
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