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Theorem sbco 2116
Description: A composition law for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
Assertion
Ref Expression
sbco  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )

Proof of Theorem sbco
StepHypRef Expression
1 sbcom3 2115 . 2  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] [ y  /  y ] ph )
2 sbid 1952 . . 3  |-  ( [ y  /  y ]
ph 
<-> 
ph )
32sbbii 1709 . 2  |-  ( [ y  /  x ] [ y  /  y ] ph  <->  [ y  /  x ] ph )
41, 3bitri 249 1  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by:  sbid2  2118  sbco3  2124  sbco3OLD  2125  sb9iOLD  2138  sb6a  2164
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