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Theorem elab2g 2760
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 2154 . 2  |-  ( A  e.  B  <->  A  e.  { x  |  ph }
)
3 elab2g.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
43elabg 2759 . 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 1289    e. wcel 1438   {cab 2074
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:  elab2  2761  elab4g  2762  eldif  3006  elun  3139  elin  3181  elsng  3456  elprg  3461  eluni  3651  eliun  3729  eliin  3730  elopab  4076  elong  4191  opeliunxp  4481  elrn2g  4614  eldmg  4619  elrnmpt  4672  elrnmpt1  4674  elimag  4765  elrnmpt2g  5739  eloprabi  5948  tfrlem3ag  6056  tfr1onlem3ag  6084  tfrcllemsucaccv  6101  elqsg  6322  isomni  6771  1idprl  7128  1idpru  7129  recexprlemell  7160  recexprlemelu  7161
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