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Theorem rabexgfGS 27224
Description: Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
Hypothesis
Ref Expression
rabexgfGS.1  |-  F/_ x A
Assertion
Ref Expression
rabexgfGS  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )

Proof of Theorem rabexgfGS
StepHypRef Expression
1 nfrab1 3047 . . . 4  |-  F/_ x { x  e.  A  |  ph }
2 rabexgfGS.1 . . . 4  |-  F/_ x A
31, 2dfss2f 3500 . . 3  |-  ( { x  e.  A  |  ph }  C_  A  <->  A. x
( x  e.  {
x  e.  A  |  ph }  ->  x  e.  A ) )
4 rabid 3043 . . . 4  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
54simplbi 460 . . 3  |-  ( x  e.  { x  e.  A  |  ph }  ->  x  e.  A )
63, 5mpgbir 1605 . 2  |-  { x  e.  A  |  ph }  C_  A
7 elex 3127 . 2  |-  ( A  e.  V  ->  A  e.  _V )
8 ssexg 4599 . 2  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  A  e.  _V )  ->  { x  e.  A  |  ph }  e.  _V )
96, 7, 8sylancr 663 1  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   F/_wnfc 2615   {crab 2821   _Vcvv 3118    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-in 3488  df-ss 3495
This theorem is referenced by:  abrexexd  27230
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