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Theorem abid2fOLD 2646
Description: Obsolete proof of abid2f 2645 as of 17-Nov-2019. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
abid2f.1  |-  F/_ x A
Assertion
Ref Expression
abid2fOLD  |-  { x  |  x  e.  A }  =  A

Proof of Theorem abid2fOLD
StepHypRef Expression
1 abid2f.1 . . . . 5  |-  F/_ x A
2 nfab1 2618 . . . . 5  |-  F/_ x { x  |  x  e.  A }
31, 2cleqf 2643 . . . 4  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } ) )
4 abid 2441 . . . . . 6  |-  ( x  e.  { x  |  x  e.  A }  <->  x  e.  A )
54bibi2i 313 . . . . 5  |-  ( ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  ( x  e.  A  <->  x  e.  A
) )
65albii 1611 . . . 4  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  x  e.  A } )  <->  A. x
( x  e.  A  <->  x  e.  A ) )
73, 6bitri 249 . . 3  |-  ( A  =  { x  |  x  e.  A }  <->  A. x ( x  e.  A  <->  x  e.  A
) )
8 biid 236 . . 3  |-  ( x  e.  A  <->  x  e.  A )
97, 8mpgbir 1596 . 2  |-  A  =  { x  |  x  e.  A }
109eqcomi 2467 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2439   F/_wnfc 2602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604
This theorem is referenced by: (None)
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