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Theorem sbcom2 1906
Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbcom2  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Distinct variable groups:    x, z    x, w    y, z
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem sbcom2
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 sbcom2v2 1905 . . . 4  |-  ( [ v  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ v  /  z ] ph )
21sbbii 1690 . . 3  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  v ] [ y  /  x ] [ v  /  z ] ph )
3 sbcom2v2 1905 . . 3  |-  ( [ w  /  v ] [ y  /  x ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  z ] ph )
42, 3bitri 182 . 2  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  z ] ph )
5 ax-17 1460 . . 3  |-  ( [ y  /  x ] ph  ->  A. v [ y  /  x ] ph )
65sbco2v 1864 . 2  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  z ] [ y  /  x ] ph )
7 ax-17 1460 . . . 4  |-  ( ph  ->  A. v ph )
87sbco2v 1864 . . 3  |-  ( [ w  /  v ] [ v  /  z ] ph  <->  [ w  /  z ] ph )
98sbbii 1690 . 2  |-  ( [ y  /  x ] [ w  /  v ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
104, 6, 93bitr3i 208 1  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] 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-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:  2sb5rf  1908  2sb6rf  1909  sbco4lem  1925  sbco4  1926  sbmo  2002  cnvopab  4776
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