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Theorem abbi 2548
Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
Assertion
Ref Expression
abbi  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )

Proof of Theorem abbi
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 hbab1 2410 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
2 hbab1 2410 . . 3  |-  ( y  e.  { x  |  ps }  ->  A. x  y  e.  { x  |  ps } )
31, 2cleqh 2532 . 2  |-  ( { x  |  ph }  =  { x  |  ps } 
<-> 
A. x ( x  e.  { x  | 
ph }  <->  x  e.  { x  |  ps }
) )
4 abid 2409 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
5 abid 2409 . . . 4  |-  ( x  e.  { x  |  ps }  <->  ps )
64, 5bibi12i 316 . . 3  |-  ( ( x  e.  { x  |  ph }  <->  x  e.  { x  |  ps }
)  <->  ( ph  <->  ps )
)
76albii 1685 . 2  |-  ( A. x ( x  e. 
{ x  |  ph } 
<->  x  e.  { x  |  ps } )  <->  A. x
( ph  <->  ps ) )
83, 7bitr2i 253 1  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435    = wceq 1437    e. wcel 1872   {cab 2407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417
This theorem is referenced by:  abbii  2551  abbid  2552  nabbi  2754  nabbiOLD  2755  rabbi  3004  sbcbi2  3348  iuneq12df  4323  dfiota2  5566  iotabi  5574  uniabio  5575  iotanul  5580  karden  8374  iuneq12daf  28172  bj-cleq  31523  abeq12  32363  elnev  36759  csbingVD  37254  csbsngVD  37263  csbxpgVD  37264  csbrngVD  37266  csbunigVD  37268  csbfv12gALTVD  37269  rabeqsn  39733
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