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Theorem sbcom4 2162
Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2163 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)
Assertion
Ref Expression
sbcom4  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Distinct variable groups:    ph, x, y, z    x, w, z
Allowed substitution hint:    ph( w)

Proof of Theorem sbcom4
StepHypRef Expression
1 nfv 1674 . . . . 5  |-  F/ z
ph
21sbf 2081 . . . 4  |-  ( [ y  /  z ]
ph 
<-> 
ph )
32sbbii 1709 . . 3  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ w  /  x ] ph )
4 nfv 1674 . . . 4  |-  F/ x ph
54sbf 2081 . . 3  |-  ( [ w  /  x ] ph 
<-> 
ph )
63, 5bitri 249 . 2  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  ph )
71sbf 2081 . . . 4  |-  ( [ w  /  z ]
ph 
<-> 
ph )
87sbbii 1709 . . 3  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  [ y  /  x ] ph )
94sbf 2081 . . 3  |-  ( [ y  /  x ] ph 
<-> 
ph )
108, 9bitri 249 . 2  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  ph )
116, 10bitr4i 252 1  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  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-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by: (None)
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