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Theorem sbco 1885
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 1711 . . 3  |-  [ y  /  x ] y  =  x
2 sbequ12 1696 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32bicomd 139 . . . 4  |-  ( y  =  x  ->  ( [ x  /  y ] ph  <->  ph ) )
43sbimi 1689 . . 3  |-  ( [ y  /  x ]
y  =  x  ->  [ y  /  x ] ( [ x  /  y ] ph  <->  ph ) )
51, 4ax-mp 7 . 2  |-  [ y  /  x ] ( [ x  /  y ] ph  <->  ph )
6 sbbi 1876 . 2  |-  ( [ y  /  x ]
( [ x  / 
y ] ph  <->  ph )  <->  ( [
y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph ) )
75, 6mpbi 143 1  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  sbco3v  1886
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