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Theorem intexrabim 4048
Description: The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexrabim  |-  ( E. x  e.  A  ph  ->  |^| { x  e.  A  |  ph }  e.  _V )

Proof of Theorem intexrabim
StepHypRef Expression
1 intexabim 4047 . 2  |-  ( E. x ( x  e.  A  /\  ph )  ->  |^| { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rex 2399 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2402 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43inteqi 3745 . . 3  |-  |^| { x  e.  A  |  ph }  =  |^| { x  |  ( x  e.  A  /\  ph ) }
54eleq1i 2183 . 2  |-  ( |^| { x  e.  A  |  ph }  e.  _V  <->  |^| { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
61, 2, 53imtr4i 200 1  |-  ( E. x  e.  A  ph  ->  |^| { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1453    e. wcel 1465   {cab 2103   E.wrex 2394   {crab 2397   _Vcvv 2660   |^|cint 3741
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-in 3047  df-ss 3054  df-int 3742
This theorem is referenced by:  cardcl  7005  isnumi  7006  cardval3ex  7009  clsval  12207
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