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Theorem bj-abeq2 31390
Description: Remove dependency on ax-13 2092 from abeq2 2561. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-abeq2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem bj-abeq2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax-5 1762 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
2 bj-hbab1 31388 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2cleqh 2553 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
4 abid 2440 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
54bibi2i 319 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
65albii 1695 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
73, 6bitri 257 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1446    = wceq 1448    e. wcel 1891   {cab 2438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-ext 2432
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448
This theorem is referenced by:  bj-abeq1  31391  bj-abbi2i  31393  bj-abbi2dv  31397  bj-clabel  31400  bj-ru1  31540
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