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Theorem sbco4lem 1986
Description: Lemma for sbco4 1987. 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 1967 . . 3  |-  ( [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ y  /  x ] [ w  /  v ] [ v  /  w ] [ w  /  y ] ph )
21sbbii 1745 . 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 1508 . . . . . . 7  |-  F/ w ph
43sbco2 1945 . . . . . 6  |-  ( [ v  /  w ] [ w  /  y ] ph  <->  [ v  /  y ] ph )
54sbbii 1745 . . . . 5  |-  ( [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ y  /  x ] [ v  /  y ] ph )
65sbbii 1745 . . . 4  |-  ( [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ w  /  v ] [ y  /  x ] [ v  /  y ] ph )
76sbbii 1745 . . 3  |-  ( [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  y ] ph )
8 nfv 1508 . . . 4  |-  F/ w [ y  /  x ] [ v  /  y ] ph
98sbco2 1945 . . 3  |-  ( [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
107, 9bitri 183 . 2  |-  ( [ x  /  w ] [ w  /  v ] [ y  /  x ] [ v  /  w ] [ w  /  y ] ph  <->  [ x  /  v ] [ y  /  x ] [ v  /  y ] ph )
11 nfv 1508 . . . . 5  |-  F/ v [ w  /  y ] ph
1211sbid2 1830 . . . 4  |-  ( [ w  /  v ] [ v  /  w ] [ w  /  y ] ph  <->  [ w  /  y ] ph )
1312sbbii 1745 . . 3  |-  ( [ y  /  x ] [ w  /  v ] [ v  /  w ] [ w  /  y ] ph  <->  [ y  /  x ] [ w  /  y ] ph )
1413sbbii 1745 . 2  |-  ( [ x  /  w ] [ y  /  x ] [ w  /  v ] [ v  /  w ] [ w  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
152, 10, 143bitr3i 209 1  |-  ( [ x  /  v ] [ y  /  x ] [ v  /  y ] ph  <->  [ x  /  w ] [ y  /  x ] [ w  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743
This theorem is referenced by:  sbco4  1987
  Copyright terms: Public domain W3C validator