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Theorem ifel 3613
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 2356 . 2  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( if (
ph ,  A ,  B )  e.  C  <->  A  e.  C ) )
2 eleq1 2356 . 2  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( if (
ph ,  A ,  B )  e.  C  <->  B  e.  C ) )
31, 2elimif 3607 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 1696   ifcif 3578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579
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