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Theorem abbi 2167
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 2050 . 2  |-  ( { x  |  ph }  =  { x  |  ps } 
<-> 
A. y ( y  e.  { x  | 
ph }  <->  y  e.  { x  |  ps }
) )
2 nfsab1 2046 . . . 4  |-  F/ x  y  e.  { x  |  ph }
3 nfsab1 2046 . . . 4  |-  F/ x  y  e.  { x  |  ps }
42, 3nfbi 1497 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )
5 nfv 1437 . . 3  |-  F/ y ( ph  <->  ps )
6 df-clab 2043 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
7 sbequ12r 1671 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
86, 7syl5bb 185 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
9 df-clab 2043 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
10 sbequ12r 1671 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ps  <->  ps ) )
119, 10syl5bb 185 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ps }  <->  ps )
)
128, 11bibi12d 228 . . 3  |-  ( y  =  x  ->  (
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )  <->  ( ph  <->  ps ) ) )
134, 5, 12cbval 1653 . 2  |-  ( A. y ( y  e. 
{ x  |  ph } 
<->  y  e.  { x  |  ps } )  <->  A. x
( ph  <->  ps ) )
141, 13bitr2i 178 1  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   [wsb 1661   {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049
This theorem is referenced by:  abbii  2169  abbid  2170  rabbi  2504  sbcbi2  2836  dfiota2  4896  iotabi  4904  uniabio  4905
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