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Theorem sbcom2v2 1998
Description: Lemma for proving sbcom2 1999. It is the same as sbcom2v 1997 but removes the distinct variable constraint on  x and  y. (Contributed by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbcom2v2  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Distinct variable groups:    x, w, z   
y, z
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem sbcom2v2
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 sbcom2v 1997 . . 3  |-  ( [ w  /  z ] [ y  /  v ] [ v  /  x ] ph  <->  [ y  /  v ] [ w  /  z ] [ v  /  x ] ph )
2 sbcom2v 1997 . . . 4  |-  ( [ w  /  z ] [ v  /  x ] ph  <->  [ v  /  x ] [ w  /  z ] ph )
32sbbii 1776 . . 3  |-  ( [ y  /  v ] [ w  /  z ] [ v  /  x ] ph  <->  [ y  /  v ] [ v  /  x ] [ w  /  z ] ph )
41, 3bitri 184 . 2  |-  ( [ w  /  z ] [ y  /  v ] [ v  /  x ] ph  <->  [ y  /  v ] [ v  /  x ] [ w  /  z ] ph )
5 ax-17 1537 . . . 4  |-  ( ph  ->  A. v ph )
65sbco2vh 1957 . . 3  |-  ( [ y  /  v ] [ v  /  x ] ph  <->  [ y  /  x ] ph )
76sbbii 1776 . 2  |-  ( [ w  /  z ] [ y  /  v ] [ v  /  x ] ph  <->  [ w  /  z ] [ y  /  x ] ph )
8 ax-17 1537 . . 3  |-  ( [ w  /  z ]
ph  ->  A. v [ w  /  z ] ph )
98sbco2vh 1957 . 2  |-  ( [ y  /  v ] [ v  /  x ] [ w  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
104, 7, 93bitr3i 210 1  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [wsb 1773
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  sbcom2  1999
  Copyright terms: Public domain W3C validator