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Theorem ifexd 4445
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 ( ph ,  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 4443 . 2  |-  ( ph  ->  if ( ph ,  A ,  B )  e.  ~P ( A  u.  B ) )
43elexd 2725 1  |-  ( ph  ->  if ( ph ,  A ,  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128   _Vcvv 2712    u. cun 3100   ifcif 3505   ~Pcpw 3543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3774
This theorem is referenced by: (None)
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