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Theorem abbi 2253
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 2133 . 2  |-  ( { x  |  ph }  =  { x  |  ps } 
<-> 
A. y ( y  e.  { x  | 
ph }  <->  y  e.  { x  |  ps }
) )
2 nfsab1 2129 . . . 4  |-  F/ x  y  e.  { x  |  ph }
3 nfsab1 2129 . . . 4  |-  F/ x  y  e.  { x  |  ps }
42, 3nfbi 1568 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )
5 nfv 1508 . . 3  |-  F/ y ( ph  <->  ps )
6 df-clab 2126 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
7 sbequ12r 1745 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
86, 7syl5bb 191 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
9 df-clab 2126 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
10 sbequ12r 1745 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ps  <->  ps ) )
119, 10syl5bb 191 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ps }  <->  ps )
)
128, 11bibi12d 234 . . 3  |-  ( y  =  x  ->  (
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )  <->  ( ph  <->  ps ) ) )
134, 5, 12cbval 1727 . 2  |-  ( A. y ( y  e. 
{ x  |  ph } 
<->  y  e.  { x  |  ps } )  <->  A. x
( ph  <->  ps ) )
141, 13bitr2i 184 1  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   [wsb 1735   {cab 2125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132
This theorem is referenced by:  abbii  2255  abbid  2256  rabbi  2608  sbcbi2  2959  dfiota2  5089  iotabi  5097  uniabio  5098
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