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

Theorem sbcomv 1981
Description: Version of sbcom 1985 with a distinct variable constraint between  x and  z. (Contributed by Jim Kingdon, 28-Feb-2018.)
Assertion
Ref Expression
sbcomv  |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
Distinct variable group:    x, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbcomv
StepHypRef Expression
1 sbco3v 1979 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] [ x  /  z ] ph )
2 sbcocom 1980 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ y  /  x ] ph )
3 sbcocom 1980 . 2  |-  ( [ y  /  x ] [ x  /  z ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
41, 2, 33bitr3i 210 1  |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [wsb 1772
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator