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Theorem sbcom2 1938
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 1937 . . . 4  |-  ( [ v  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ v  /  z ] ph )
21sbbii 1721 . . 3  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  v ] [ y  /  x ] [ v  /  z ] ph )
3 sbcom2v2 1937 . . 3  |-  ( [ w  /  v ] [ y  /  x ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  z ] ph )
42, 3bitri 183 . 2  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  z ] ph )
5 ax-17 1489 . . 3  |-  ( [ y  /  x ] ph  ->  A. v [ y  /  x ] ph )
65sbco2v 1896 . 2  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  z ] [ y  /  x ] ph )
7 ax-17 1489 . . . 4  |-  ( ph  ->  A. v ph )
87sbco2v 1896 . . 3  |-  ( [ w  /  v ] [ v  /  z ] ph  <->  [ w  /  z ] ph )
98sbbii 1721 . 2  |-  ( [ y  /  x ] [ w  /  v ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
104, 6, 93bitr3i 209 1  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  2sb5rf  1940  2sb6rf  1941  sbco4lem  1957  sbco4  1958  sbmo  2034  cnvopab  4898
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