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Theorem intexab 4595
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 3803 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 intex 4593 . 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 1617    e. wcel 1823   {cab 2439    =/= wne 2649   _Vcvv 3106   (/)c0 3783   |^|cint 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-int 4272
This theorem is referenced by:  intexrab  4596  tcmin  8163  cfval  8618  efgval  16934
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