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Theorem sbco4 2186
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 2158 . . 3  |-  ( [ x  /  v ] [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph )
2 nfv 1674 . . . . 5  |-  F/ u [ v  /  y ] ph
32sbco2 2118 . . . 4  |-  ( [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ y  /  x ] [ v  /  y ] ph )
43sbbii 1709 . . 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 2185 . 2  |-  ( [ x  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  t ] [ y  /  x ] [ t  /  y ] ph )
7 sbco4lem 2185 . 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 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by: (None)
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