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Theorem ifel 3775
Description: Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
Assertion
Ref Expression
ifel  |-  ( if ( ph ,  A ,  B )  e.  C  <->  ( ( ph  /\  A  e.  C )  \/  ( -.  ph  /\  B  e.  C ) ) )

Proof of Theorem ifel
StepHypRef Expression
1 eleq1 2497 . 2  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( if (
ph ,  A ,  B )  e.  C  <->  A  e.  C ) )
2 eleq1 2497 . 2  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( if (
ph ,  A ,  B )  e.  C  <->  B  e.  C ) )
31, 2elimif 3769 1  |-  ( if ( ph ,  A ,  B )  e.  C  <->  ( ( ph  /\  A  e.  C )  \/  ( -.  ph  /\  B  e.  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    /\ wa 360    e. wcel 1726   ifcif 3740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
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 2424  df-cleq 2430  df-clel 2433  df-if 3741
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