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Theorem elrabd 2771
Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2769. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
elrabd.1  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
elrabd.2  |-  ( ph  ->  A  e.  B )
elrabd.3  |-  ( ph  ->  ch )
Assertion
Ref Expression
elrabd  |-  ( ph  ->  A  e.  { x  e.  B  |  ps } )
Distinct variable groups:    x, A    x, B    ch, x
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem elrabd
StepHypRef Expression
1 elrabd.2 . . 3  |-  ( ph  ->  A  e.  B )
2 elrabd.3 . . 3  |-  ( ph  ->  ch )
31, 2jca 300 . 2  |-  ( ph  ->  ( A  e.  B  /\  ch ) )
4 elrabd.1 . . 3  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
54elrab 2769 . 2  |-  ( A  e.  { x  e.  B  |  ps }  <->  ( A  e.  B  /\  ch ) )
63, 5sylibr 132 1  |-  ( ph  ->  A  e.  { x  e.  B  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {crab 2363
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-rab 2368  df-v 2621
This theorem is referenced by:  znnen  11293
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