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Theorem intexab 4551
Description: The intersection of a nonempty 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 3757 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 intex 4549 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  |^| { x  | 
ph }  e.  _V )
31, 2bitr3i 251 1  |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1587    e. wcel 1758   {cab 2436    =/= wne 2644   _Vcvv 3071   (/)c0 3738   |^|cint 4229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-int 4230
This theorem is referenced by:  intexrab  4552  tcmin  8065  cfval  8520  efgval  16327
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