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Theorem abid2f 2457
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 2434 . . . . 5  |-  F/_ x { x  |  x  e.  A }
31, 2cleqf 2456 . . . 4  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } ) )
4 abid 2284 . . . . . 6  |-  ( x  e.  { x  |  x  e.  A }  <->  x  e.  A )
54bibi2i 304 . . . . 5  |-  ( ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  ( x  e.  A  <->  x  e.  A
) )
65albii 1556 . . . 4  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  A. x
( x  e.  A  <->  x  e.  A ) )
73, 6bitri 240 . . 3  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  A
) )
8 biid 227 . . 3  |-  ( x  e.  A  <->  x  e.  A )
97, 8mpgbir 1540 . 2  |-  A  =  { x  |  x  e.  A }
109eqcomi 2300 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419
This theorem is referenced by:  mptctf  23363  rabexgf  27798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421
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