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Theorem abid2f 2445
Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
abid2f.1  |-  F/_ x A
Assertion
Ref Expression
abid2f  |-  { x  |  x  e.  A }  =  A

Proof of Theorem abid2f
StepHypRef Expression
1 abid2f.1 . . . . 5  |-  F/_ x A
2 nfab1 2422 . . . . 5  |-  F/_ x { x  |  x  e.  A }
31, 2cleqf 2444 . . . 4  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } ) )
4 abid 2272 . . . . . 6  |-  ( x  e.  { x  |  x  e.  A }  <->  x  e.  A )
54bibi2i 306 . . . . 5  |-  ( ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  ( x  e.  A  <->  x  e.  A
) )
65albii 1554 . . . 4  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  A. x
( x  e.  A  <->  x  e.  A ) )
73, 6bitri 242 . . 3  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  A
) )
8 biid 229 . . 3  |-  ( x  e.  A  <->  x  e.  A )
97, 8mpgbir 1538 . 2  |-  A  =  { x  |  x  e.  A }
109eqcomi 2288 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1528    = wceq 1624    e. wcel 1685   {cab 2270   F/_wnfc 2407
This theorem is referenced by:  rabexgf  27094
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409
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