ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abbi Unicode version

Theorem abbi 2151
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 2034 . 2  |-  ( { x  |  ph }  =  { x  |  ps } 
<-> 
A. y ( y  e.  { x  | 
ph }  <->  y  e.  { x  |  ps }
) )
2 nfsab1 2030 . . . 4  |-  F/ x  y  e.  { x  |  ph }
3 nfsab1 2030 . . . 4  |-  F/ x  y  e.  { x  |  ps }
42, 3nfbi 1481 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )
5 nfv 1421 . . 3  |-  F/ y ( ph  <->  ps )
6 df-clab 2027 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
7 sbequ12r 1655 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
86, 7syl5bb 181 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ph }  <->  ph ) )
9 df-clab 2027 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
10 sbequ12r 1655 . . . . 5  |-  ( y  =  x  ->  ( [ y  /  x ] ps  <->  ps ) )
119, 10syl5bb 181 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x  |  ps }  <->  ps )
)
128, 11bibi12d 224 . . 3  |-  ( y  =  x  ->  (
( y  e.  {
x  |  ph }  <->  y  e.  { x  |  ps } )  <->  ( ph  <->  ps ) ) )
134, 5, 12cbval 1637 . 2  |-  ( A. y ( y  e. 
{ x  |  ph } 
<->  y  e.  { x  |  ps } )  <->  A. x
( ph  <->  ps ) )
141, 13bitr2i 174 1  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   [wsb 1645   {cab 2026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033
This theorem is referenced by:  abbii  2153  abbid  2154  rabbi  2487  dfiota2  4868  iotabi  4876  uniabio  4877
  Copyright terms: Public domain W3C validator