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Theorem intexab 4185
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 3486 . 2  |-  ( { x  |  ph }  =/=  (/)  <->  E. x ph )
2 intex 4183 . 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 1531    e. wcel 1696   {cab 2282    =/= wne 2459   _Vcvv 2801   (/)c0 3468   |^|cint 3878
This theorem is referenced by:  intexrab  4186  tcmin  7442  cfval  7889  efgval  15042
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-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-int 3879
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