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Theorem abbi 2576
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 2451 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
2 hbab1 2451 . . 3  |-  ( y  e.  { x  |  ps }  ->  A. x  y  e.  { x  |  ps } )
31, 2cleqh 2563 . 2  |-  ( { x  |  ph }  =  { x  |  ps } 
<-> 
A. x ( x  e.  { x  | 
ph }  <->  x  e.  { x  |  ps }
) )
4 abid 2450 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
5 abid 2450 . . . 4  |-  ( x  e.  { x  |  ps }  <->  ps )
64, 5bibi12i 321 . . 3  |-  ( ( x  e.  { x  |  ph }  <->  x  e.  { x  |  ps }
)  <->  ( ph  <->  ps )
)
76albii 1702 . 2  |-  ( A. x ( x  e. 
{ x  |  ph } 
<->  x  e.  { x  |  ps } )  <->  A. x
( ph  <->  ps ) )
83, 7bitr2i 258 1  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1453    = wceq 1455    e. wcel 1898   {cab 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458
This theorem is referenced by:  abbii  2578  abbid  2579  nabbi  2737  rabbi  2981  sbcbi2  3328  rabeqsn  4013  iuneq12df  4316  dfiota2  5570  iotabi  5578  uniabio  5579  iotanul  5584  karden  8397  iuneq12daf  28225  bj-cleq  31601  abeq12  32445  elnev  36834  csbingVD  37322  csbsngVD  37331  csbxpgVD  37332  csbrngVD  37334  csbunigVD  37336  csbfv12gALTVD  37337
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