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Theorem ifnmfalse 27935
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 3661 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 2532 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 iffalse 3661 . 2  |-  ( -.  A  e.  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
31, 2sylbi 187 1  |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1647    e. wcel 1715    e/ wnel 2530   ifcif 3654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nel 2532  df-if 3655
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