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Theorem sbco2d 2027
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 1821 . . . 4  |-  F/ z ( ph  ->  ps )
43sbco2 2026 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  x ] ( ph  ->  ps ) )
5 sbco2d.1 . . . . . 6  |-  F/ x ph
65sbrim 2007 . . . . 5  |-  ( [ z  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ z  /  x ] ps ) )
76sbbii 1634 . . . 4  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) )
81sbrim 2007 . . . 4  |-  ( [ y  /  z ] ( ph  ->  [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  z ] [
z  /  x ] ps ) )
97, 8bitri 240 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  z ] [ z  /  x ] ps ) )
105sbrim 2007 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
114, 9, 103bitr3i 266 . 2  |-  ( (
ph  ->  [ y  / 
z ] [ z  /  x ] ps ) 
<->  ( ph  ->  [ y  /  x ] ps ) )
1211pm5.74ri 237 1  |-  ( ph  ->  ( [ y  / 
z ] [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531   [wsb 1629
This theorem is referenced by:  sbco3  2028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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