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Theorem intexrab 4361
Description: The intersection of a non-empty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
Assertion
Ref Expression
intexrab  |-  ( E. x  e.  A  ph  <->  |^|
{ x  e.  A  |  ph }  e.  _V )

Proof of Theorem intexrab
StepHypRef Expression
1 intexab 4360 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  |^|
{ x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rex 2713 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2716 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43inteqi 4056 . . 3  |-  |^| { x  e.  A  |  ph }  =  |^| { x  |  ( x  e.  A  /\  ph ) }
54eleq1i 2501 . 2  |-  ( |^| { x  e.  A  |  ph }  e.  _V  <->  |^| { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
61, 2, 53bitr4i 270 1  |-  ( E. x  e.  A  ph  <->  |^|
{ x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    e. wcel 1726   {cab 2424   E.wrex 2708   {crab 2711   _Vcvv 2958   |^|cint 4052
This theorem is referenced by:  onintrab2  4784  rankf  7722  rankvalb  7725  cardf2  7832  tskmval  8716  lspval  16053  aspval  16389  clsval  17103  spanval  22837  rgspnval  27352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-int 4053
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