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Theorem abid2f 2599
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 2576 . . . . 5  |-  F/_ x { x  |  x  e.  A }
31, 2cleqf 2598 . . . 4  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } ) )
4 abid 2426 . . . . . 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 1576 . . . 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 1560 . 2  |-  A  =  { x  |  x  e.  A }
109eqcomi 2442 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2424   F/_wnfc 2561
This theorem is referenced by:  mptctf  24114  rabexgf  27673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563
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