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Theorem ifpr 3622
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 2748 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2748 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 ifcl 3542 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  _V )
4 ifeqor 3543 . . . 4  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
5 elprg 3598 . . . 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 226 . . 3  |-  ( if ( ph ,  A ,  B )  e.  _V  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
73, 6syl 17 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
81, 2, 7syl2an 465 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740   ifcif 3506   {cpr 3582
This theorem is referenced by:  suppr  7152  uvcvvcl  26568
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-if 3507  df-sn 3587  df-pr 3588
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