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Theorem elrabd 2813
Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2811. (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 302 . 2  |-  ( ph  ->  ( A  e.  B  /\  ch ) )
4 elrabd.1 . . 3  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
54elrab 2811 . 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 1314    e. wcel 1463   {crab 2395
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660
This theorem is referenced by:  ctssdccl  6962  suplocexprlemru  7491  suplocexprlemloc  7493  znnen  11806  ennnfonelemj0  11809  ennnfonelemg  11811  cdivcncfap  12651  cnopnap  12658  limcdifap  12683  limcimolemlt  12685  dvcoapbr  12723  subctctexmid  13007
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