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Theorem ifexd 4513
Description: Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
Hypotheses
Ref Expression
ifexd.1  |-  ( ph  ->  A  e.  V )
ifexd.2  |-  ( ph  ->  B  e.  W )
Assertion
Ref Expression
ifexd  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  _V )

Proof of Theorem ifexd
StepHypRef Expression
1 ifexd.1 . . 3  |-  ( ph  ->  A  e.  V )
2 ifexd.2 . . 3  |-  ( ph  ->  B  e.  W )
31, 2ifelpwund 4511 . 2  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  ~P ( A  u.  B ) )
43elexd 2773 1  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2164   _Vcvv 2760    u. cun 3151   ifcif 3557   ~Pcpw 3601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pr 4238  ax-un 4462
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836
This theorem is referenced by:  gsumfzval  12964
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