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

Theorem cbvabw 2263
Description: Version of cbvab 2264 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvabw.1  |-  F/ y
ph
cbvabw.2  |-  F/ x ps
cbvabw.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvabw  |-  { x  |  ph }  =  {
y  |  ps }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvabw
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvabw.1 . . . . 5  |-  F/ y
ph
21sbco2v 1922 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
3 cbvabw.2 . . . . . 6  |-  F/ x ps
4 cbvabw.3 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
53, 4sbiev 1766 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
65sbbii 1739 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] ps )
72, 6bitr3i 185 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
8 df-clab 2127 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
9 df-clab 2127 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
107, 8, 93bitr4i 211 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1110eqriv 2137 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1332   F/wnf 1437    e. wcel 1481   [wsb 1736   {cab 2126
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133
This theorem is referenced by:  cbvsbcw  2939
  Copyright terms: Public domain W3C validator