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Theorem elrabd 2870
Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2868. (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 304 . 2  |-  ( ph  ->  ( A  e.  B  /\  ch ) )
4 elrabd.1 . . 3  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
54elrab 2868 . 2  |-  ( A  e.  { x  e.  B  |  ps }  <->  ( A  e.  B  /\  ch ) )
63, 5sylibr 133 1  |-  ( ph  ->  A  e.  { x  e.  B  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   {crab 2439
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714
This theorem is referenced by:  ctssdccl  7050  suplocexprlemru  7634  suplocexprlemloc  7636  phisum  12107  znnen  12114  ennnfonelemj0  12117  ennnfonelemg  12119  cdivcncfap  12974  cnopnap  12981  ivthinc  13008  limcdifap  13018  limcimolemlt  13020  dvcoapbr  13058  subctctexmid  13560
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