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Theorem sbco2d 2173
Description: A composition law for substitution. (Contributed by NM, 2-Jun-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 1937 . . . 4  |-  F/ z ( ph  ->  ps )
43sbco2 2172 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  x ] ( ph  ->  ps ) )
5 sbco2d.1 . . . . . 6  |-  F/ x ph
65sbrim 2151 . . . . 5  |-  ( [ z  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ z  /  x ] ps ) )
76sbbii 1764 . . . 4  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) )
81sbrim 2151 . . . 4  |-  ( [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  z ] [
z  /  x ] ps ) )
97, 8bitri 249 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  z ] [ z  /  x ] ps ) )
105sbrim 2151 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
114, 9, 103bitr3i 275 . 2  |-  ( (
ph  ->  [ y  / 
z ] [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  x ] ps ) )
1211pm5.74ri 246 1  |-  ( ph  ->  ( [ y  / 
z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   F/wnf 1631   [wsb 1757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1628  df-nf 1632  df-sb 1758
This theorem is referenced by:  sbco3  2174
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