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Theorem sbco 2023
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 1975 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12 1860 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32bicomd 192 . . . 4  |-  ( y  =  x  ->  ( [ x  /  y ] ph  <->  ph ) )
43sbimi 1633 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ x  /  y ] ph  <->  ph ) )
51, 4ax-mp 8 . 2  |-  [ y  /  x ] ( [ x  /  y ] ph  <->  ph )
6 sbbi 2011 . 2  |-  ( [ y  /  x ]
( [ x  / 
y ] ph  <->  ph )  <->  ( [
y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph ) )
75, 6mpbi 199 1  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1629
This theorem is referenced by:  sbid2  2024  sbco3  2028  sb9i  2034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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