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Theorem ifpr 3694
Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
Assertion
Ref Expression
ifpr  |-  ( ( A  e.  C  /\  B  e.  D )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )

Proof of Theorem ifpr
StepHypRef Expression
1 elex 2809 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2809 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 ifcl 3614 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  _V )
4 ifeqor 3615 . . . 4  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
5 elprg 3670 . . . 4  |-  ( if ( ph ,  A ,  B )  e.  _V  ->  ( if ( ph ,  A ,  B )  e.  { A ,  B }  <->  ( if (
ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B ) ) )
64, 5mpbiri 224 . . 3  |-  ( if ( ph ,  A ,  B )  e.  _V  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
73, 6syl 15 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
81, 2, 7syl2an 463 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   ifcif 3578   {cpr 3654
This theorem is referenced by:  suppr  7235  indf  23614  uvcvvcl  27339
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-nfc 2421  df-v 2803  df-un 3170  df-if 3579  df-sn 3659  df-pr 3660
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