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Theorem elabg 2852
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 2385 . 2  |-  F/_ x A
2 nfv 1629 . 2  |-  F/ x ps
3 elabg.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 2849 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 2239
This theorem is referenced by:  elab2g  2853  intmin3  3788  finds  4573  elxpi  4612  elfi  7051  inficl  7062  dffi3  7068  scott0  7440  elgch  8124  nqpr  8518  hashf1lem1  11270  efgcpbllemb  14899  frgpuplem  14916  lspsn  15594  eltg  16527  eltg2  16528  fbssfi  17364  mpfind  19260  pf1ind  19270  eloi  24251  elixp2b  24320  domrancur1b  24366  domrancur1c  24368  isoriso2  24379  isdir2  24458  intopcoaconlem3b  24704  iscol2  25259  islocfin  25462  setindtrs  26284  rngunsnply  26544  islshpkrN  28214
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729
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