Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rabexgf Unicode version

Theorem rabexgf 27798
Description: A version of rabexg 4180 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 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 simpl 443 . . . . 5  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
32ss2abi 3258 . . . 4  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  { x  |  x  e.  A }
4 rabexgf.1 . . . . 5  |-  F/_ x A
54abid2f 2457 . . . 4  |-  { x  |  x  e.  A }  =  A
63, 5sseqtri 3223 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
71, 6eqsstri 3221 . 2  |-  { x  e.  A  |  ph }  C_  A
8 ssexg 4176 . 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 1696   {cab 2282   F/_wnfc 2419   {crab 2560   _Vcvv 2801    C_ wss 3165
This theorem is referenced by:  stoweidlem27  27879  stoweidlem35  27887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179
  Copyright terms: Public domain W3C validator