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Theorem elab3 2921
Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
Hypotheses
Ref Expression
elab3.1  |-  ( ps 
->  A  e.  _V )
elab3.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elab3  |-  ( A  e.  { x  | 
ph }  <->  ps )
Distinct variable groups:    ps, x    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem elab3
StepHypRef Expression
1 elab3.1 . 2  |-  ( ps 
->  A  e.  _V )
2 elab3.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32elab3g 2920 . 2  |-  ( ( ps  ->  A  e.  _V )  ->  ( A  e.  { x  | 
ph }  <->  ps )
)
41, 3ax-mp 8 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788
This theorem is referenced by:  fvelrnb  5570  elrnmpt2  5957  ovelrn  5996  isfi  6885  isnum2  7578  pm54.43lem  7632  isfin3  7922  isfin5  7925  isfin6  7926  genpelv  8624  iswrd  11415  4sqlem2  12996  vdwapval  13020  ismnd  14369  isghm  14683  issrng  15615  lspsnel  15760  lspprel  15847  iscss  16583  istps  16674  islp  16872  is2ndc  17172  elpt  17267  itg2l  19084  elply  19577  vtarsuelt  25895  ellspd  27254  rngunsnply  27378  isline  29928  ispointN  29931  ispsubsp  29934  ispsubclN  30126  islaut  30272  ispautN  30288  istendo  30949
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
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-v 2790
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