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Theorem rabid2 3039
Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rabid2  |-  ( A  =  { x  e.  A  |  ph }  <->  A. x  e.  A  ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rabid2
StepHypRef Expression
1 abeq2 2591 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
2 pm4.71 630 . . . 4  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  ph )
) )
32albii 1620 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
41, 3bitr4i 252 . 2  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  ->  ph ) )
5 df-rab 2823 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65eqeq2i 2485 . 2  |-  ( A  =  { x  e.  A  |  ph }  <->  A  =  { x  |  ( x  e.  A  /\  ph ) } )
7 df-ral 2819 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
84, 6, 73bitr4i 277 1  |-  ( A  =  { x  e.  A  |  ph }  <->  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2819  df-rab 2823
This theorem is referenced by:  rabxm  3808  iinrab2  4388  riinrab  4401  class2seteq  4615  dmmptg  5504  dmmptd  5711  fneqeql  5990  fmpt  6043  zfrep6  6753  axdc2lem  8829  ioomax  11600  iccmax  11601  hashbc  12469  dfphi2  14166  phiprmpw  14168  isnsg4  16058  symggen2  16311  psgnfvalfi  16353  lssuni  17398  psr1baslem  18035  psgnghm2  18424  ocv0  18515  dsmmfi  18576  frlmfibas  18602  frlmlbs  18638  ordtrest2lem  19510  comppfsc  19860  xkouni  19927  xkoccn  19947  tsmsfbas  20453  clsocv  21517  ehlbase  21665  ovolicc2lem4  21758  itg2monolem1  21984  musum  23292  lgsquadlem2  23455  vdgr1d  24676  vdgr1b  24677  frgraregorufr0  24826  ubthlem1  25559  xrsclat  27427  ordtrest2NEWlem  27655  hasheuni  27842  measvuni  27936  ddemeas  27959  imambfm  27984  subfacp1lem6  28380  conpcon  28431  cvmliftmolem2  28478  cvmlift2lem12  28510  tfisg  29137  wfisg  29142  frinsg  29178  fdc  30068  isbnd3  30110  vdioph  30544  fiphp3d  30584  phisum  30991  stirlinglem14  31614  pmap1N  34780  pol1N  34923  dia1N  36067  dihwN  36303
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