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Theorem sbcom2 1998
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 1997 . . . 4  |-  ( [ v  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ v  /  z ] ph )
21sbbii 1775 . . 3  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  v ] [ y  /  x ] [ v  /  z ] ph )
3 sbcom2v2 1997 . . 3  |-  ( [ w  /  v ] [ y  /  x ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  z ] ph )
42, 3bitri 184 . 2  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  z ] ph )
5 ax-17 1536 . . 3  |-  ( [ y  /  x ] ph  ->  A. v [ y  /  x ] ph )
65sbco2vh 1956 . 2  |-  ( [ w  /  v ] [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  z ] [ y  /  x ] ph )
7 ax-17 1536 . . . 4  |-  ( ph  ->  A. v ph )
87sbco2vh 1956 . . 3  |-  ( [ w  /  v ] [ v  /  z ] ph  <->  [ w  /  z ] ph )
98sbbii 1775 . 2  |-  ( [ y  /  x ] [ w  /  v ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
104, 6, 93bitr3i 210 1  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773
This theorem is referenced by:  2sb5rf  2000  2sb6rf  2001  sbco4lem  2017  sbco4  2018  sbmo  2096  cnvopab  5044
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