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Theorem elab2 2805
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
Hypotheses
Ref Expression
elab2.1  |-  A  e. 
_V
elab2.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
elab2.3  |-  B  =  { x  |  ph }
Assertion
Ref Expression
elab2  |-  ( A  e.  B  <->  ps )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elab2
StepHypRef Expression
1 elab2.1 . 2  |-  A  e. 
_V
2 elab2.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 elab2.3 . . 3  |-  B  =  { x  |  ph }
42, 3elab2g 2804 . 2  |-  ( A  e.  _V  ->  ( A  e.  B  <->  ps )
)
51, 4ax-mp 5 1  |-  ( A  e.  B  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  elpw  3486  elint  3747  opabid  4149  elrn2  4751  elimasn  4876  oprabid  5771  tfrlem3a  6175  tfrcllemsucaccv  6219  tfrcllembxssdm  6221  tfrcllemres  6227  addnqprlemrl  7333  addnqprlemru  7334  addnqprlemfl  7335  addnqprlemfu  7336  mulnqprlemrl  7349  mulnqprlemru  7350  mulnqprlemfl  7351  mulnqprlemfu  7352  ltnqpr  7369  ltnqpri  7370  archpr  7419  cauappcvgprlemladdfu  7430  cauappcvgprlemladdfl  7431  caucvgprlemladdfu  7453  caucvgprprlemopu  7475  suplocexprlemloc  7497  txuni2  12352
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