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Theorem sbcocom 1844
Description: Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)
Assertion
Ref Expression
sbcocom  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ z  /  x ] ph )

Proof of Theorem sbcocom
StepHypRef Expression
1 equsb1 1668 . . 3  |-  [ z  /  y ] y  =  z
2 sbequ 1721 . . . 4  |-  ( y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
32sbimi 1647 . . 3  |-  ( [ z  /  y ] y  =  z  ->  [ z  /  y ] ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
41, 3ax-mp 7 . 2  |-  [ z  /  y ] ( [ y  /  x ] ph  <->  [ z  /  x ] ph )
5 sbbi 1833 . 2  |-  ( [ z  /  y ] ( [ y  /  x ] ph  <->  [ z  /  x ] ph )  <->  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ z  /  x ] ph ) )
64, 5mpbi 133 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ z  /  x ] 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-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:  sbcomv  1845  sbco3xzyz  1847  sbcom  1849
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