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Theorem bj-abeq2 32597
Description: Remove dependency on ax-13 1952 from abeq2 2575. (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 1671 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
2 bj-hbab1 32594 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2bj-cleqh 32596 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
4 abid 2438 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
54bibi2i 313 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
65albii 1611 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
73, 6bitri 249 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2436
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-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446
This theorem is referenced by:  bj-abeq1  32598  bj-abbi2i  32600  bj-abbi2dv  32604  bj-clabel  32607  bj-ru1  32739
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