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Theorem sbcom 2123
Description: A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-2018.)
Assertion
Ref Expression
sbcom  |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )

Proof of Theorem sbcom
StepHypRef Expression
1 sbco3 2121 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] [ x  /  z ] ph )
2 sbcom3 2112 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ y  /  x ] ph )
3 sbcom3 2112 . 2  |-  ( [ y  /  x ] [ x  /  z ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
41, 2, 33bitr3i 275 1  |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] 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:  wl-sbcom3  28551
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