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Theorem bj-rabtrALT 32736
Description: Alternate proof of bj-rabtr 32735. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-rabtrALT  |-  { x  e.  A  | T.  }  =  A
Distinct variable group:    x, A

Proof of Theorem bj-rabtrALT
StepHypRef Expression
1 nfrab1 3000 . . 3  |-  F/_ x { x  e.  A  | T.  }
2 nfcv 2613 . . 3  |-  F/_ x A
31, 2cleqf 2639 . 2  |-  ( { x  e.  A  | T.  }  =  A  <->  A. x
( x  e.  {
x  e.  A  | T.  }  <->  x  e.  A
) )
4 tru 1374 . . 3  |- T.
5 rabid 2996 . . 3  |-  ( x  e.  { x  e.  A  | T.  }  <->  ( x  e.  A  /\ T.  ) )
64, 5mpbiran2 910 . 2  |-  ( x  e.  { x  e.  A  | T.  }  <->  x  e.  A )
73, 6mpgbir 1596 1  |-  { x  e.  A  | T.  }  =  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370   T. wtru 1371    e. wcel 1758   {crab 2799
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 1952  ax-ext 2430
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 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804
This theorem is referenced by: (None)
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