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Theorem sbco3v 1981
Description: Version of sbco3 1986 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 1951 . . . 4  |-  F/ x [ y  /  x ] ph
21nfri 1530 . . 3  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
32sbco2vh 1957 . 2  |-  ( [ z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ y  /  x ] ph )
4 sbco 1980 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  [ x  /  y ] ph )
54sbbii 1776 . 2  |-  ( [ z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
63, 5bitr3i 186 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] 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:  sbcomv  1983
  Copyright terms: Public domain W3C validator