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Theorem intexrab 4186
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 4185 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  |^|
{ x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rex 2562 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2565 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43inteqi 3882 . . 3  |-  |^| { x  e.  A  |  ph }  =  |^| { x  |  ( x  e.  A  /\  ph ) }
54eleq1i 2359 . 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 1531    e. wcel 1696   {cab 2282   E.wrex 2557   {crab 2560   _Vcvv 2801   |^|cint 3878
This theorem is referenced by:  onintrab2  4609  rankf  7482  rankvalb  7485  cardf2  7592  tskmval  8477  lspval  15748  aspval  16084  clsval  16790  spanval  21928  rgspnval  27476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-int 3879
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