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Theorem rabid2 2640
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 2273 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
2 pm4.71 387 . . . 4  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  ph )
) )
32albii 1457 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
41, 3bitr4i 186 . 2  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  ->  ph ) )
5 df-rab 2451 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65eqeq2i 2175 . 2  |-  ( A  =  { x  e.  A  |  ph }  <->  A  =  { x  |  ( x  e.  A  /\  ph ) } )
7 df-ral 2447 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
84, 6, 73bitr4i 211 1  |-  ( A  =  { x  e.  A  |  ph }  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1340    = wceq 1342    e. wcel 2135   {cab 2150   A.wral 2442   {crab 2446
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-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-ral 2447  df-rab 2451
This theorem is referenced by:  rabxmdc  3436  rabrsndc  3639  class2seteq  4137  dmmptg  5096  dmmptd  5313  fneqeql  5588  fmpt  5630  acexmidlemph  5830  inffiexmid  6864  ssfirab  6891  exmidaclem  7156  ioomax  9876  iccmax  9877  dfphi2  12141  phiprmpw  12143  phisum  12161  unennn  12293  znnen  12294
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