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Theorem ifpr 3848
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 2956 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2956 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 ifcl 3767 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  _V )
4 ifeqor 3768 . . . 4  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
5 elprg 3823 . . . 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 1652    e. wcel 1725   _Vcvv 2948   ifcif 3731   {cpr 3807
This theorem is referenced by:  suppr  7462  indf  24401  uvcvvcl  27151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-if 3732  df-sn 3812  df-pr 3813
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