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Theorem sb8ab 2201
Description: Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
Hypothesis
Ref Expression
sb8ab.1  |-  F/ y
ph
Assertion
Ref Expression
sb8ab  |-  { x  |  ph }  =  {
y  |  [ y  /  x ] ph }

Proof of Theorem sb8ab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sb8ab.1 . . . 4  |-  F/ y
ph
21sbco2 1881 . . 3  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
3 df-clab 2069 . . 3  |-  ( z  e.  { y  |  [ y  /  x ] ph }  <->  [ z  /  y ] [
y  /  x ] ph )
4 df-clab 2069 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
52, 3, 43bitr4ri 211 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  [ y  /  x ] ph } )
65eqriv 2079 1  |-  { x  |  ph }  =  {
y  |  [ y  /  x ] ph }
Colors of variables: wff set class
Syntax hints:    = 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: (None)
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