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Theorem elrab3 2814
Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
Hypothesis
Ref Expression
elrab.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elrab3  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  ph }  <->  ps ) )
Distinct variable groups:    ps, x    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem elrab3
StepHypRef Expression
1 elrab.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21elrab 2813 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
32baib 889 1  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  ph }  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   {crab 2397
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662
This theorem is referenced by:  unimax  3740  undifexmid  4087  frind  4244  ordtriexmidlem2  4406  ordtriexmid  4407  ordtri2orexmid  4408  onsucelsucexmid  4415  0elsucexmid  4450  ordpwsucexmid  4455  ordtri2or2exmid  4456  acexmidlema  5733  acexmidlemb  5734  isnumi  7006  genpelvl  7288  genpelvu  7289  cauappcvgprlemladdru  7432  cauappcvgprlem1  7435  caucvgprlem1  7455  sup3exmid  8683  supinfneg  9358  infsupneg  9359  supminfex  9360  ublbneg  9373  negm  9375  hashinfuni  10491  infssuzex  11569  gcddvds  11579  dvdslegcd  11580  bezoutlemsup  11624  lcmval  11671  dvdslcm  11677  isprm2lem  11724
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