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Theorem elrab3 2918
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 2917 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
32baib 920 1  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  ph }  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2164   {crab 2476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762
This theorem is referenced by:  unimax  3870  undifexmid  4223  frind  4384  ordtriexmidlem2  4553  ordtriexmid  4554  ontriexmidim  4555  ordtri2orexmid  4556  onsucelsucexmid  4563  0elsucexmid  4598  ordpwsucexmid  4603  ordtri2or2exmid  4604  ontri2orexmidim  4605  canth  5872  acexmidlema  5910  acexmidlemb  5911  isnumi  7244  genpelvl  7574  genpelvu  7575  cauappcvgprlemladdru  7718  cauappcvgprlem1  7721  caucvgprlem1  7741  sup3exmid  8978  supinfneg  9663  infsupneg  9664  supminfex  9665  ublbneg  9681  negm  9683  hashinfuni  10851  infssuzex  12089  gcddvds  12103  dvdslegcd  12104  bezoutlemsup  12149  uzwodc  12177  lcmval  12204  dvdslcm  12210  isprm2lem  12257
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