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Theorem pm11.07 2190
Description: (Probably not) Axiom *11.07 in [WhiteheadRussell] p. 159. The original confusingly reads: *11.07 "Whatever possible argument  x may be,  ph ( x ,  y ) is true whatever possible argument  y may be" implies the corresponding statement with  x and  y interchanged except in " ph ( x ,  y )". This theorem will be deleted after 22-Feb-2018 if no one is able to determine the correct interpretation. See https://groups.google.com/d/msg/metamath/iS0fOvSemC8/YzrRyX70AgAJ. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.) (New usage is discouraged.)
Assertion
Ref Expression
pm11.07  |-  ( [ 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 pm11.07
StepHypRef Expression
1 nfv 1629 . . . . 5  |-  F/ z
ph
21sbf 2105 . . . 4  |-  ( [ y  /  z ]
ph 
<-> 
ph )
32sbbii 1665 . . 3  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ w  /  x ] ph )
4 nfv 1629 . . . 4  |-  F/ x ph
54sbf 2105 . . 3  |-  ( [ w  /  x ] ph 
<-> 
ph )
63, 5bitri 241 . 2  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  ph )
71sbf 2105 . . . 4  |-  ( [ w  /  z ]
ph 
<-> 
ph )
87sbbii 1665 . . 3  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  [ y  /  x ] ph )
94sbf 2105 . . 3  |-  ( [ y  /  x ] ph 
<-> 
ph )
108, 9bitri 241 . 2  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  ph )
116, 10bitr4i 244 1  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659
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