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Theorem cbvab 2202
Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
cbvab.1  |-  F/ y
ph
cbvab.2  |-  F/ x ps
cbvab.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvab  |-  { x  |  ph }  =  {
y  |  ps }

Proof of Theorem cbvab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvab.2 . . . . 5  |-  F/ x ps
21nfsb 1864 . . . 4  |-  F/ x [ z  /  y ] ps
3 cbvab.1 . . . . . 6  |-  F/ y
ph
4 cbvab.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
54equcoms 1635 . . . . . . 7  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
65bicomd 139 . . . . . 6  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
73, 6sbie 1715 . . . . 5  |-  ( [ x  /  y ] ps  <->  ph )
8 sbequ 1762 . . . . 5  |-  ( x  =  z  ->  ( [ x  /  y ] ps  <->  [ z  /  y ] ps ) )
97, 8syl5bbr 192 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  y ] ps ) )
102, 9sbie 1715 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
11 df-clab 2069 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
12 df-clab 2069 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
1310, 11, 123bitr4i 210 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1413eqriv 2079 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   F/wnf 1390    e. wcel 1434   [wsb 1686   {cab 2068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075
This theorem is referenced by:  cbvabv  2203  cbvrab  2600  cbvsbc  2843  cbvrabcsf  2968  dfdmf  4556  dfrnf  4603  funfvdm2f  5270  abrexex2g  5778  abrexex2  5782
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