MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbco4 Structured version   Unicode version

Theorem sbco4 2212
Description: Two ways of exchanging two variables. Both sides of the biconditional exchange  x and  y, either via two temporary variables  u and  v, or a single temporary  w. (Contributed by Jim Kingdon, 25-Sep-2018.)
Assertion
Ref Expression
sbco4  |-  ( [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
Distinct variable groups:    v, u, ph    x, u, v    y, u, v    ph, w    x, w    y, w
Allowed substitution hints:    ph( x, y)

Proof of Theorem sbco4
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 sbcom2 2191 . . 3  |-  ( [ x  /  v ] [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph )
2 nfv 1712 . . . . 5  |-  F/ u [ v  /  y ] ph
32sbco2 2160 . . . 4  |-  ( [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ y  /  x ] [ v  /  y ] ph )
43sbbii 1751 . . 3  |-  ( [ x  /  v ] [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
51, 4bitr3i 251 . 2  |-  ( [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
6 sbco4lem 2211 . 2  |-  ( [ x  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  t ] [ y  /  x ] [ t  /  y ] ph )
7 sbco4lem 2211 . 2  |-  ( [ x  /  t ] [ y  /  x ] [ t  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
85, 6, 73bitri 271 1  |-  ( [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator