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Theorem sbco2d 2164
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
sbco2d.1  |-  F/ x ph
sbco2d.2  |-  F/ z
ph
sbco2d.3  |-  ( ph  ->  F/ 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  |-  F/ z
ph
2 sbco2d.3 . . . . 5  |-  ( ph  ->  F/ z ps )
31, 2nfim1 1831 . . . 4  |-  F/ z ( ph  ->  ps )
43sbco2 2163 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  x ] ( ph  ->  ps ) )
5 sbco2d.1 . . . . . 6  |-  F/ x ph
65sbrim 2138 . . . . 5  |-  ( [ z  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ z  /  x ] ps ) )
76sbbii 1666 . . . 4  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) )
81sbrim 2138 . . . 4  |-  ( [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  z ] [
z  /  x ] ps ) )
97, 8bitri 242 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  z ] [ z  /  x ] ps ) )
105sbrim 2138 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
114, 9, 103bitr3i 268 . 2  |-  ( (
ph  ->  [ y  / 
z ] [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  x ] ps ) )
1211pm5.74ri 239 1  |-  ( ph  ->  ( [ y  / 
z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   F/wnf 1554   [wsb 1659
This theorem is referenced by:  sbco3  2165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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