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Theorem sbco2d 1966
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 1570 . . . 4  |-  ( (
ph  ->  ps )  ->  A. z ( ph  ->  ps ) )
43sbco2h 1964 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  x ] ( ph  ->  ps ) )
5 sbco2d.1 . . . . . 6  |-  ( ph  ->  A. x ph )
65sbrim 1956 . . . . 5  |-  ( [ z  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ z  /  x ] ps ) )
76sbbii 1765 . . . 4  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) )
81sbrim 1956 . . . 4  |-  ( [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  z ] [
z  /  x ] ps ) )
97, 8bitri 184 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  z ] [ z  /  x ] ps ) )
105sbrim 1956 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
114, 9, 103bitr3i 210 . 2  |-  ( (
ph  ->  [ y  / 
z ] [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  x ] ps ) )
1211pm5.74ri 181 1  |-  ( ph  ->  ( [ y  / 
z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351   [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by: (None)
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