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Theorem ifnmfalse 27245
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 3513 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 2422 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 iffalse 3513 . 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 5    -> wi 6    = wceq 1619    e. wcel 1621    e/ wnel 2420   ifcif 3506
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nel 2422  df-if 3507
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