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Theorem rabexgf 27695
Description: A version of rabexg 4164 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 2552 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 simpl 443 . . . . 5  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
32ss2abi 3245 . . . 4  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  x  e.  A }
4 rabexgf.1 . . . . 5  |-  F/_ x A
54abid2f 2444 . . . 4  |-  { x  |  x  e.  A }  =  A
63, 5sseqtri 3210 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
71, 6eqsstri 3208 . 2  |-  { x  e.  A  |  ph }  C_  A
8 ssexg 4160 . 2  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  A  e.  V
)  ->  { x  e.  A  |  ph }  e.  _V )
97, 8mpan 651 1  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   {cab 2269   F/_wnfc 2406   {crab 2547   _Vcvv 2788    C_ wss 3152
This theorem is referenced by:  stoweidlem27  27776  stoweidlem35  27784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166
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