ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  equsb3 Unicode version

Theorem equsb3 1967
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
Assertion
Ref Expression
equsb3  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
Distinct variable group:    x, z

Proof of Theorem equsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 1966 . . 3  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
21sbbii 1776 . 2  |-  ( [ y  /  w ] [ w  /  x ] x  =  z  <->  [ y  /  w ]
w  =  z )
3 ax-17 1537 . . 3  |-  ( x  =  z  ->  A. w  x  =  z )
43sbco2vh 1961 . 2  |-  ( [ y  /  w ] [ w  /  x ] x  =  z  <->  [ y  /  x ]
x  =  z )
5 equsb3lem 1966 . 2  |-  ( [ y  /  w ]
w  =  z  <->  y  =  z )
62, 4, 53bitr3i 210 1  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  sb8eu  2055  sb8euh  2065  sb8iota  5222
  Copyright terms: Public domain W3C validator