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Theorem elif 3716
Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
Assertion
Ref Expression
elif  |-  ( A  e.  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  e.  B )  \/  ( -.  ph  /\  A  e.  C )
) )

Proof of Theorem elif
StepHypRef Expression
1 eleq2 2448 . 2  |-  ( if ( ph ,  B ,  C )  =  B  ->  ( A  e.  if ( ph ,  B ,  C )  <->  A  e.  B ) )
2 eleq2 2448 . 2  |-  ( if ( ph ,  B ,  C )  =  C  ->  ( A  e.  if ( ph ,  B ,  C )  <->  A  e.  C ) )
31, 2elimif 3711 1  |-  ( A  e.  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  e.  B )  \/  ( -.  ph  /\  A  e.  C )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    e. wcel 1717   ifcif 3682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-if 3683
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