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Theorem bj-rabtrALT 34899
Description: Alternate proof of bj-rabtr 34898. (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 3035 . . 3  |-  F/_ x { x  e.  A  | T.  }
2 nfcv 2616 . . 3  |-  F/_ x A
31, 2cleqf 2643 . 2  |-  ( { x  e.  A  | T.  }  =  A  <->  A. x
( x  e.  {
x  e.  A  | T.  }  <->  x  e.  A
) )
4 tru 1402 . . 3  |- T.
5 rabid 3031 . . 3  |-  ( x  e.  { x  e.  A  | T.  }  <->  ( x  e.  A  /\ T.  ) )
64, 5mpbiran2 917 . 2  |-  ( x  e.  { x  e.  A  | T.  }  <->  x  e.  A )
73, 6mpgbir 1627 1  |-  { x  e.  A  | T.  }  =  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398   T. wtru 1399    e. wcel 1823   {crab 2808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813
This theorem is referenced by: (None)
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