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Theorem abbi 2393
Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
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 dfcleq 2277 . 2  |-  ( { x  |  ph }  =  { x  |  ps } 
<-> 
A. y ( y  e.  { x  | 
ph }  <->  y  e.  { x  |  ps }
) )
2 nfsab1 2273 . . . 4  |-  F/ x  y  e.  { x  |  ph }
3 nfsab1 2273 . . . 4  |-  F/ x  y  e.  { x  |  ps }
42, 3nfbi 1772 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )
5 nfv 1605 . . 3  |-  F/ y ( ph  <->  ps )
6 df-clab 2270 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
7 sbequ12r 1861 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
86, 7syl5bb 248 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
9 df-clab 2270 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
10 sbequ12r 1861 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ps  <->  ps ) )
119, 10syl5bb 248 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ps }  <->  ps )
)
128, 11bibi12d 312 . . 3  |-  ( y  =  x  ->  (
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )  <->  ( ph  <->  ps ) ) )
134, 5, 12cbval 1924 . 2  |-  ( A. y ( y  e. 
{ x  |  ph } 
<->  y  e.  { x  |  ps } )  <->  A. x
( ph  <->  ps ) )
141, 13bitr2i 241 1  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269
This theorem is referenced by:  abbii  2395  abbid  2396  rabbi  2718  dfiota2  5220  iotabi  5228  uniabio  5229  iotanul  5234  karden  7565  iuneq12daf  23154  iuneq12df  23155  rabbidva2  23164  elnev  27638  csbingVD  28660  csbsngVD  28669  csbxpgVD  28670  csbrngVD  28672  csbunigVD  28674  csbfv12gALTVD  28675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276
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