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Theorem rabexgf 31560
Description: A version of rabexg 4606 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 2816 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 simpl 457 . . . . 5  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
32ss2abi 3568 . . . 4  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  x  e.  A }
4 rabexgf.1 . . . . 5  |-  F/_ x A
54abid2f 2648 . . . 4  |-  { x  |  x  e.  A }  =  A
63, 5sseqtri 3531 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
71, 6eqsstri 3529 . 2  |-  { x  e.  A  |  ph }  C_  A
8 ssexg 4602 . 2  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  A  e.  V
)  ->  { x  e.  A  |  ph }  e.  _V )
97, 8mpan 670 1  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   {cab 2442   F/_wnfc 2605   {crab 2811   _Vcvv 3109    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-in 3478  df-ss 3485
This theorem is referenced by:  stoweidlem27  31970  stoweidlem35  31978
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