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Theorem sbco 2022
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbco  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )

Proof of Theorem sbco
StepHypRef Expression
1 equsb2 1980 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12 1861 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32bicomd 194 . . . 4  |-  ( y  =  x  ->  ( [ x  /  y ] ph  <->  ph ) )
43sbimi 1635 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ x  /  y ] ph  <->  ph ) )
51, 4ax-mp 10 . 2  |-  [ y  /  x ] ( [ x  /  y ] ph  <->  ph )
6 sbbi 2010 . 2  |-  ( [ y  /  x ]
( [ x  / 
y ] ph  <->  ph )  <->  ( [
y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph ) )
75, 6mpbi 201 1  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   [wsb 1631
This theorem is referenced by:  sbid2  2023  sbco3  2027  sb9i  2033
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632
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