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Theorem sbco 2132
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 2084 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12 1940 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32bicomd 193 . . . 4  |-  ( y  =  x  ->  ( [ x  /  y ] ph  <->  ph ) )
43sbimi 1660 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ x  /  y ] ph  <->  ph ) )
51, 4ax-mp 8 . 2  |-  [ y  /  x ] ( [ x  /  y ] ph  <->  ph )
6 sbbi 2120 . 2  |-  ( [ y  /  x ]
( [ x  / 
y ] ph  <->  ph )  <->  ( [
y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph ) )
75, 6mpbi 200 1  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1655
This theorem is referenced by:  sbid2  2133  sbco3  2137  sb9i  2143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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