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Theorem sbco4 1926
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 1906 . . 3  |-  ( [ x  /  v ] [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph )
2 nfv 1462 . . . . 5  |-  F/ u [ v  /  y ] ph
32sbco2 1882 . . . 4  |-  ( [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ y  /  x ] [ v  /  y ] ph )
43sbbii 1690 . . 3  |-  ( [ x  /  v ] [ y  /  u ] [ u  /  x ] [ v  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
51, 4bitr3i 184 . 2  |-  ( [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
6 sbco4lem 1925 . 2  |-  ( [ x  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  t ] [ y  /  x ] [ t  /  y ] ph )
7 sbco4lem 1925 . 2  |-  ( [ x  /  t ] [ y  /  x ] [ t  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
85, 6, 73bitri 204 1  |-  ( [ y  /  u ] [ x  /  v ] [ u  /  x ] [ v  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by: (None)
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