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Theorem ifpr 3992
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 3043 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 3043 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 ifcl 3899 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  _V )
4 ifeqor 3901 . . . 4  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
5 elprg 3960 . . . 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 233 . . 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 475 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   ifcif 3857   {cpr 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-un 3394  df-if 3858  df-sn 3945  df-pr 3947
This theorem is referenced by:  suppr  7844  uvcvvcl  18907  indf  28164
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