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Theorem sbcom 2163
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 2162 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] [ x  /  z ] ph )
2 sbcom3 2155 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ y  /  x ] ph )
3 sbcom3 2155 . 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 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745
This theorem is referenced by:  wl-sbcom3  30275
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