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Theorem bj-abbid 30928
Description: Remove dependency on ax-13 2026 from abbid 2537. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-abbid.1  |-  F/ x ph
bj-abbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
bj-abbid  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )

Proof of Theorem bj-abbid
StepHypRef Expression
1 bj-abbid.1 . . 3  |-  F/ x ph
2 bj-abbid.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2alrimi 1901 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 bj-abbi 30925 . 2  |-  ( A. x ( ps  <->  ch )  <->  { x  |  ps }  =  { x  |  ch } )
53, 4sylib 196 1  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1403    = wceq 1405   F/wnf 1637   {cab 2387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394
This theorem is referenced by:  bj-abbidv  30929
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