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Theorem ifpr 3816
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 2924 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2924 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 ifcl 3735 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  _V )
4 ifeqor 3736 . . . 4  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
5 elprg 3791 . . . 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 225 . . 3  |-  ( if ( ph ,  A ,  B )  e.  _V  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
73, 6syl 16 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
81, 2, 7syl2an 464 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 358    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   ifcif 3699   {cpr 3775
This theorem is referenced by:  suppr  7429  indf  24366  uvcvvcl  27104
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-if 3700  df-sn 3780  df-pr 3781
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