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Theorem ifelpwung 4442
Description: Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
Assertion
Ref Expression
ifelpwung  |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  ~P ( A  u.  B ) )

Proof of Theorem ifelpwung
StepHypRef Expression
1 ifssun 3519 . 2  |-  if (
ph ,  A ,  B )  C_  ( A  u.  B )
2 unexg 4404 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  e.  _V )
3 elpw2g 4118 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  ( if ( ph ,  A ,  B )  e.  ~P ( A  u.  B
)  <->  if ( ph ,  A ,  B )  C_  ( A  u.  B
) ) )
42, 3syl 14 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( if ( ph ,  A ,  B )  e.  ~P ( A  u.  B )  <->  if ( ph ,  A ,  B )  C_  ( A  u.  B )
) )
51, 4mpbiri 167 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  ~P ( A  u.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 2128   _Vcvv 2712    u. cun 3100    C_ wss 3102   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:  ifelpwund  4443  ifelpwun  4444
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