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Theorem ifnmfalse 28244
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 3574 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 2451 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 iffalse 3574 . 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 1625    e. wcel 1686    e/ wnel 2449   ifcif 3567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nel 2451  df-if 3568
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