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Theorem sbco 1978
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 1908 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12 1893 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32bicomd 194 . . . 4  |-  ( y  =  x  ->  ( [ x  /  y ] ph  <->  ph ) )
43sbimi 1885 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ x  /  y ] ph  <->  ph ) )
51, 4ax-mp 10 . 2  |-  [ y  /  x ] ( [ x  /  y ] ph  <->  ph )
6 sbbi 1964 . 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 1883
This theorem is referenced by:  sbid2  1979  sbco3  1983  sb9i  1989
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
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