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Theorem elab2g 2834
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 2207 . 2  |-  ( A  e.  B  <->  A  e.  { x  |  ph }
)
3 elab2g.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
43elabg 2833 . 2  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
52, 4syl5bb 191 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1332    e. wcel 1481   {cab 2126
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691
This theorem is referenced by:  elab2  2835  elab4g  2836  eldif  3083  elun  3220  elin  3262  elsng  3545  elprg  3550  eluni  3745  eliun  3823  eliin  3824  elopab  4186  elong  4301  opeliunxp  4600  elrn2g  4735  eldmg  4740  elrnmpt  4794  elrnmpt1  4796  elimag  4891  elrnmpog  5889  eloprabi  6100  tfrlem3ag  6212  tfr1onlem3ag  6240  tfrcllemsucaccv  6257  elqsg  6485  elixp2  6602  isomni  7014  ismkv  7033  iswomni  7045  1idprl  7420  1idpru  7421  recexprlemell  7452  recexprlemelu  7453  mertenslemub  11333  mertenslemi1  11334  mertenslem2  11335  istopg  12198  isbasisg  12243
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