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Theorem intexrab 4170
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 4169 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  |^|
{ x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rex 2549 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2552 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43inteqi 3866 . . 3  |-  |^| { x  e.  A  |  ph }  =  |^| { x  |  ( x  e.  A  /\  ph ) }
54eleq1i 2346 . 2  |-  ( |^| { x  e.  A  |  ph }  e.  _V  <->  |^| { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
61, 2, 53bitr4i 268 1  |-  ( E. x  e.  A  ph  <->  |^|
{ x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   {cab 2269   E.wrex 2544   {crab 2547   _Vcvv 2788   |^|cint 3862
This theorem is referenced by:  onintrab2  4593  rankf  7466  rankvalb  7469  cardf2  7576  tskmval  8461  lspval  15732  aspval  16068  clsval  16774  spanval  21912  rgspnval  27373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-int 3863
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