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

Theorem vjust 2731
Description: Soundness justification theorem for df-v 2732. (Contributed by Rodolfo Medina, 27-Apr-2010.)
Assertion
Ref Expression
vjust  |-  { x  |  x  =  x }  =  { y  |  y  =  y }

Proof of Theorem vjust
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 equid 1694 . . . . 5  |-  x  =  x
21sbt 1777 . . . 4  |-  [ z  /  x ] x  =  x
3 equid 1694 . . . . 5  |-  y  =  y
43sbt 1777 . . . 4  |-  [ z  /  y ] y  =  y
52, 42th 173 . . 3  |-  ( [ z  /  x ]
x  =  x  <->  [ z  /  y ] y  =  y )
6 df-clab 2157 . . 3  |-  ( z  e.  { x  |  x  =  x }  <->  [ z  /  x ]
x  =  x )
7 df-clab 2157 . . 3  |-  ( z  e.  { y  |  y  =  y }  <->  [ z  /  y ] y  =  y )
85, 6, 73bitr4i 211 . 2  |-  ( z  e.  { x  |  x  =  x }  <->  z  e.  { y  |  y  =  y } )
98eqriv 2167 1  |-  { x  |  x  =  x }  =  { y  |  y  =  y }
Colors of variables: wff set class
Syntax hints:    = wceq 1348   [wsb 1755    e. wcel 2141   {cab 2156
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-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator