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Theorem elab4g 2764
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
Hypotheses
Ref Expression
elab4g.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
elab4g.2  |-  B  =  { x  |  ph }
Assertion
Ref Expression
elab4g  |-  ( A  e.  B  <->  ( A  e.  _V  /\  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elab4g
StepHypRef Expression
1 elex 2630 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 elab4g.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 elab4g.2 . . 3  |-  B  =  { x  |  ph }
42, 3elab2g 2762 . 2  |-  ( A  e.  _V  ->  ( A  e.  B  <->  ps )
)
51, 4biadan2 444 1  |-  ( A  e.  B  <->  ( A  e.  _V  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> 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: (None)
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