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Theorem rabexgf 27626
Description: A version of rabexg 4345 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypothesis
Ref Expression
rabexgf.1  |-  F/_ x A
Assertion
Ref Expression
rabexgf  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )

Proof of Theorem rabexgf
StepHypRef Expression
1 df-rab 2706 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 simpl 444 . . . . 5  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
32ss2abi 3407 . . . 4  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  x  e.  A }
4 rabexgf.1 . . . . 5  |-  F/_ x A
54abid2f 2596 . . . 4  |-  { x  |  x  e.  A }  =  A
63, 5sseqtri 3372 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
71, 6eqsstri 3370 . 2  |-  { x  e.  A  |  ph }  C_  A
8 ssexg 4341 . 2  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  A  e.  V
)  ->  { x  e.  A  |  ph }  e.  _V )
97, 8mpan 652 1  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   {cab 2421   F/_wnfc 2558   {crab 2701   _Vcvv 2948    C_ wss 3312
This theorem is referenced by:  stoweidlem27  27707  stoweidlem35  27715
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-in 3319  df-ss 3326
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