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Theorem sbcom 1849
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 1848 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] [ x  /  z ] ph )
2 sbcocom 1844 . 2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ y  /  x ] ph )
3 sbcocom 1844 . 2  |-  ( [ y  /  x ] [ x  /  z ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
41, 2, 33bitr3i 199 1  |-  ( [ y  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ y  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 98   [wsb 1645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646
This theorem is referenced by:  sb9  1855
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