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Theorem intexab 4358
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 3646 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 intex 4356 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  |^| { x  | 
ph }  e.  _V )
31, 2bitr3i 243 1  |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    e. wcel 1725   {cab 2422    =/= wne 2599   _Vcvv 2956   (/)c0 3628   |^|cint 4050
This theorem is referenced by:  intexrab  4359  tcmin  7680  cfval  8127  efgval  15349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-int 4051
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