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Theorem ifel 3600
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 2343 . 2  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( if (
ph ,  A ,  B )  e.  C  <->  A  e.  C ) )
2 eleq1 2343 . 2  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( if (
ph ,  A ,  B )  e.  C  <->  B  e.  C ) )
31, 2elimif 3594 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 176    \/ wo 357    /\ wa 358    e. wcel 1684   ifcif 3565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-if 3566
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