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Theorem intexab 4169
Description: The intersection of a non-empty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
Assertion
Ref Expression
intexab  |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )

Proof of Theorem intexab
StepHypRef Expression
1 abn0 3473 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 intex 4167 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  |^| { x  | 
ph }  e.  _V )
31, 2bitr3i 242 1  |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    e. wcel 1684   {cab 2269    =/= wne 2446   _Vcvv 2788   (/)c0 3455   |^|cint 3862
This theorem is referenced by:  intexrab  4170  tcmin  7426  cfval  7873  efgval  15026
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-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-int 3863
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