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Theorem rabid2 2486
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 2146 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
2 pm4.71 369 . . . 4  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  ph )
) )
32albii 1359 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
41, 3bitr4i 176 . 2  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  ->  ph ) )
5 df-rab 2315 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65eqeq2i 2050 . 2  |-  ( A  =  { x  e.  A  |  ph }  <->  A  =  { x  |  ( x  e.  A  /\  ph ) } )
7 df-ral 2311 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
84, 6, 73bitr4i 201 1  |-  ( A  =  { x  e.  A  |  ph }  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2306   {crab 2310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rab 2315
This theorem is referenced by:  rabxmdc  3249  rabrsndc  3438  class2seteq  3916  dmmptg  4818  fneqeql  5275  fmpt  5319  acexmidlemph  5505  ioomax  8817  iccmax  8818
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