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Theorem elab3 3032
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 3031 . 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 177    = wceq 1649    e. wcel 1717   {cab 2373   _Vcvv 2899
This theorem is referenced by:  fvelrnb  5713  elrnmpt2  6122  ovelrn  6161  isfi  7067  isnum2  7765  pm54.43lem  7819  isfin3  8109  isfin5  8112  isfin6  8113  genpelv  8810  iswrd  11656  4sqlem2  13244  vdwapval  13268  ismnd  14619  isghm  14933  issrng  15865  lspsnel  16006  lspprel  16093  iscss  16833  istps  16924  islp  17127  is2ndc  17430  elpt  17525  itg2l  19488  elply  19981  ellspd  26923  rngunsnply  27047  isline  29853  ispointN  29856  ispsubsp  29859  ispsubclN  30051  islaut  30197  ispautN  30213  istendo  30874
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 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901
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