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Theorem sbco2d 1940
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
sbco2d.1  |-  ( ph  ->  A. x ph )
sbco2d.2  |-  ( ph  ->  A. z ph )
sbco2d.3  |-  ( ph  ->  ( ps  ->  A. z ps ) )
Assertion
Ref Expression
sbco2d  |-  ( ph  ->  ( [ y  / 
z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )

Proof of Theorem sbco2d
StepHypRef Expression
1 sbco2d.2 . . . . 5  |-  ( ph  ->  A. z ph )
2 sbco2d.3 . . . . 5  |-  ( ph  ->  ( ps  ->  A. z ps ) )
31, 2hbim1 1550 . . . 4  |-  ( (
ph  ->  ps )  ->  A. z ( ph  ->  ps ) )
43sbco2h 1938 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  x ] ( ph  ->  ps ) )
5 sbco2d.1 . . . . . 6  |-  ( ph  ->  A. x ph )
65sbrim 1930 . . . . 5  |-  ( [ z  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ z  /  x ] ps ) )
76sbbii 1739 . . . 4  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) )
81sbrim 1930 . . . 4  |-  ( [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  z ] [
z  /  x ] ps ) )
97, 8bitri 183 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  z ] [ z  /  x ] ps ) )
105sbrim 1930 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
114, 9, 103bitr3i 209 . 2  |-  ( (
ph  ->  [ y  / 
z ] [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  x ] ps ) )
1211pm5.74ri 180 1  |-  ( ph  ->  ( [ y  / 
z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330   [wsb 1736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737
This theorem is referenced by: (None)
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