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Theorem sbco3v 1885
Description: Version of sbco3 1890 with a distinct variable constraint between  x and  y. (Contributed by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbco3v  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbco3v
StepHypRef Expression
1 nfs1v 1857 . . . 4  |-  F/ x [ y  /  x ] ph
21nfri 1453 . . 3  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
32sbco2v 1863 . 2  |-  ( [ z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ y  /  x ] ph )
4 sbco 1884 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  [ x  /  y ] ph )
54sbbii 1689 . 2  |-  ( [ z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
63, 5bitr3i 184 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  sbcomv  1887
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