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Theorem sbco2yz 1885
Description: This is a version of sbco2 1887 where  z is distinct from 
y. It is a lemma on the way to proving sbco2 1887 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
sbco2yz.1  |-  F/ z
ph
Assertion
Ref Expression
sbco2yz  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbco2yz
StepHypRef Expression
1 sbco2yz.1 . . . 4  |-  F/ z
ph
21nfsb 1870 . . 3  |-  F/ z [ y  /  x ] ph
32nfri 1457 . 2  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
4 sbequ 1768 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
53, 4sbieh 1720 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   F/wnf 1394   [wsb 1692
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sbco2h  1886
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