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Theorem sbcom 1987
Description: A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
Assertion
Ref Expression
sbcom  |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )

Proof of Theorem sbcom
StepHypRef Expression
1 sbco3 1986 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] [ x  /  z ] ph )
2 sbcocom 1982 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ y  /  x ] ph )
3 sbcocom 1982 . 2  |-  ( [ y  /  x ] [ x  /  z ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
41, 2, 33bitr3i 210 1  |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] 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-bndl 1520  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:  sb9  1991
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