ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8ab Unicode version

Theorem sb8ab 2259
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 1936 . . 3  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
3 df-clab 2124 . . 3  |-  ( z  e.  { y  |  [ y  /  x ] ph }  <->  [ z  /  y ] [
y  /  x ] ph )
4 df-clab 2124 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
52, 3, 43bitr4ri 212 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  [ y  /  x ] ph } )
65eqriv 2134 1  |-  { x  |  ph }  =  {
y  |  [ y  /  x ] ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1331   F/wnf 1436    e. wcel 1480   [wsb 1735   {cab 2123
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator