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Theorem sbco3xzyz 1946
Description: Version of sbco3 1947 with distinct variable constraints between  x and  z, and  y and  z. Lemma for proving sbco3 1947. (Contributed by Jim Kingdon, 22-Mar-2018.)
Assertion
Ref Expression
sbco3xzyz  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbco3xzyz
StepHypRef Expression
1 sbcomxyyz 1945 . 2  |-  ( [ z  /  y ] [ z  /  x ] ph  <->  [ z  /  x ] [ z  /  y ] ph )
2 sbcocom 1943 . 2  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ z  /  x ] ph )
3 sbcocom 1943 . 2  |-  ( [ z  /  x ] [ x  /  y ] ph  <->  [ z  /  x ] [ z  /  y ] ph )
41, 2, 33bitr4i 211 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1735
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sbco3  1947
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