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Theorem elabg 2890
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
Hypothesis
Ref Expression
elabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elabg  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem elabg
StepHypRef Expression
1 nfcv 2394 . 2  |-  F/_ x A
2 nfv 1629 . 2  |-  F/ x ps
3 elabg.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 2887 1  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   {cab 2244
This theorem is referenced by:  elab2g  2891  intmin3  3864  finds  4654  elxpi  4693  elfi  7135  inficl  7146  dffi3  7152  scott0  7524  elgch  8212  nqpr  8606  hashf1lem1  11358  efgcpbllemb  15026  frgpuplem  15043  lspsn  15721  eltg  16657  eltg2  16658  fbssfi  17494  mpfind  19390  pf1ind  19400  elabreximd  23000  ballotlemfmpn  23014  eloi  24452  elixp2b  24521  domrancur1b  24567  domrancur1c  24569  isoriso2  24580  isdir2  24659  intopcoaconlem3b  24905  iscol2  25460  islocfin  25663  setindtrs  26485  rngunsnply  26745  islshpkrN  28477
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765
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