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Theorem elab2g 2741
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
Hypotheses
Ref Expression
elab2g.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
elab2g.2  |-  B  =  { x  |  ph }
Assertion
Ref Expression
elab2g  |-  ( A  e.  V  ->  ( A  e.  B  <->  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)    V( x)

Proof of Theorem elab2g
StepHypRef Expression
1 elab2g.2 . . 3  |-  B  =  { x  |  ph }
21eleq2i 2146 . 2  |-  ( A  e.  B  <->  A  e.  { x  |  ph }
)
3 elab2g.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
43elabg 2740 . 2  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
52, 4syl5bb 190 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2068
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604
This theorem is referenced by:  elab2  2742  elab4g  2743  eldif  2983  elun  3114  elin  3156  elsng  3415  elprg  3420  eluni  3606  eliun  3684  eliin  3685  elopab  4015  elong  4130  opeliunxp  4415  elrn2g  4547  eldmg  4552  elrnmpt  4605  elrnmpt1  4607  elimag  4696  elrnmpt2g  5638  eloprabi  5847  tfrlem3ag  5952  tfr1onlem3ag  5980  tfrcllemsucaccv  5997  elqsg  6215  1idprl  6831  1idpru  6832  recexprlemell  6863  recexprlemelu  6864
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