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Theorem abid2f 2306
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 2283 . . . . 5  |-  F/_ x { x  |  x  e.  A }
31, 2cleqf 2305 . . . 4  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } ) )
4 abid 2127 . . . . . 6  |-  ( x  e.  { x  |  x  e.  A }  <->  x  e.  A )
54bibi2i 226 . . . . 5  |-  ( ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  ( x  e.  A  <->  x  e.  A
) )
65albii 1446 . . . 4  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  A. x
( x  e.  A  <->  x  e.  A ) )
73, 6bitri 183 . . 3  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  A
) )
8 biid 170 . . 3  |-  ( x  e.  A  <->  x  e.  A )
97, 8mpgbir 1429 . 2  |-  A  =  { x  |  x  e.  A }
109eqcomi 2143 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   {cab 2125   F/_wnfc 2268
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270
This theorem is referenced by: (None)
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