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Theorem elrab3 2722
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 2721 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
32baib 839 1  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  ph }  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576
This theorem is referenced by:  unimax  3642  frind  4117  ordtriexmidlem2  4274  ordtriexmid  4275  ordtri2orexmid  4276  onsucelsucexmid  4283  0elsucexmid  4317  ordpwsucexmid  4322  ordtri2or2exmid  4324  acexmidlema  5531  acexmidlemb  5532  isnumi  6420  genpelvl  6668  genpelvu  6669  cauappcvgprlemladdru  6812  cauappcvgprlem1  6815  caucvgprlem1  6835  ublbneg  8645  negm  8647
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