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Theorem sbco2yz 1914
Description: This is a version of sbco2 1916 where  z is distinct from 
y. It is a lemma on the way to proving sbco2 1916 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 1899 . . 3  |-  F/ z [ y  /  x ] ph
32nfri 1484 . 2  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
4 sbequ 1796 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
53, 4sbieh 1748 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   F/wnf 1421   [wsb 1720
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by:  sbco2h  1915
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