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Theorem sbco4lem 2185
Description: Lemma for sbco4 2186. It replaces the temporary variable  v with another temporary variable  w. (Contributed by Jim Kingdon, 26-Sep-2018.)
Assertion
Ref Expression
sbco4lem  |-  ( [ x  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
Distinct variable groups:    w, v, ph    x, v, w    y,
v, w
Allowed substitution hints:    ph( x, y)

Proof of Theorem sbco4lem
StepHypRef Expression
1 sbcom2 2158 . . 3  |-  ( [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  w ] [ w  /  y ] ph )
21sbbii 1709 . 2  |-  ( [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  v ] [ v  /  w ] [ w  /  y ] ph )
3 nfv 1674 . . . . . . 7  |-  F/ w ph
43sbco2 2118 . . . . . 6  |-  ( [ v  /  w ] [ w  /  y ] ph  <->  [ v  /  y ] ph )
54sbbii 1709 . . . . 5  |-  ( [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ y  /  x ] [ v  /  y ] ph )
65sbbii 1709 . . . 4  |-  ( [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ w  /  v ] [ y  /  x ] [ v  /  y ] ph )
76sbbii 1709 . . 3  |-  ( [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  y ] ph )
8 nfv 1674 . . . 4  |-  F/ w [ y  /  x ] [ v  /  y ] ph
98sbco2 2118 . . 3  |-  ( [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
107, 9bitri 249 . 2  |-  ( [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
11 nfv 1674 . . . . 5  |-  F/ v [ w  /  y ] ph
1211sbid2 2115 . . . 4  |-  ( [ w  /  v ] [ v  /  w ] [ w  /  y ] ph  <->  [ w  /  y ] ph )
1312sbbii 1709 . . 3  |-  ( [ y  /  x ] [ w  /  v ] [ v  /  w ] [ w  /  y ] ph  <->  [ y  /  x ] [ w  /  y ] ph )
1413sbbii 1709 . 2  |-  ( [ x  /  w ] [ y  /  x ] [ w  /  v ] [ v  /  w ] [ w  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
152, 10, 143bitr3i 275 1  |-  ( [ x  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] 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:  sbco4  2186
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