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Theorem elab 2758
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
Hypotheses
Ref Expression
elab.1  |-  A  e. 
_V
elab.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elab  |-  ( A  e.  { x  | 
ph }  <->  ps )
Distinct variable groups:    ps, x    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem elab
StepHypRef Expression
1 nfv 1466 . 2  |-  F/ x ps
2 elab.1 . 2  |-  A  e. 
_V
3 elab.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabf 2757 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  ralab  2773  rexab  2775  intab  3712  dfiin2g  3758  dfiunv2  3761  uniuni  4264  dcextest  4386  peano5  4403  finds  4405  finds2  4406  funcnvuni  5069  fun11iun  5258  elabrex  5519  abrexco  5520  mapval2  6415  ssenen  6547  snexxph  6638  sbthlem2  6646  indpi  6880  nqprm  7080  nqprrnd  7081  nqprdisj  7082  nqprloc  7083  nqprl  7089  nqpru  7090  cauappcvgprlem2  7198  caucvgprlem2  7218  peano1nnnn  7368  peano2nnnn  7369  1nn  8405  peano2nn  8406  dfuzi  8826  hashfacen  10206  shftfvalg  10217  ovshftex  10218  shftfval  10220  bj-ssom  11488
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