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Theorem rabexgf 27365
Description: A version of rabexg 4296 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 2660 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 simpl 444 . . . . 5  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
32ss2abi 3360 . . . 4  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  x  e.  A }
4 rabexgf.1 . . . . 5  |-  F/_ x A
54abid2f 2550 . . . 4  |-  { x  |  x  e.  A }  =  A
63, 5sseqtri 3325 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
71, 6eqsstri 3323 . 2  |-  { x  e.  A  |  ph }  C_  A
8 ssexg 4292 . 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 1717   {cab 2375   F/_wnfc 2512   {crab 2655   _Vcvv 2901    C_ wss 3265
This theorem is referenced by:  stoweidlem27  27446  stoweidlem35  27454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rab 2660  df-v 2903  df-in 3272  df-ss 3279
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