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Theorem sbco3 2028
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbco3  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )

Proof of Theorem sbco3
StepHypRef Expression
1 drsb1 1962 . . 3  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  y ] [
y  /  x ] ph ) )
2 sbequ12a 1862 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
32alimi 1546 . . . 4  |-  ( A. x  x  =  y  ->  A. x ( [ y  /  x ] ph 
<->  [ x  /  y ] ph ) )
4 spsbbi 2017 . . . 4  |-  ( A. x ( [ y  /  x ] ph  <->  [ x  /  y ]
ph )  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
53, 4syl 15 . . 3  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
61, 5bitr3d 246 . 2  |-  ( A. x  x  =  y  ->  ( [ z  / 
y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
7 sbco 2023 . . . 4  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  [ x  /  y ] ph )
87sbbii 1634 . . 3  |-  ( [ z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
9 nfnae 1896 . . . 4  |-  F/ y  -.  A. x  x  =  y
10 nfnae 1896 . . . 4  |-  F/ x  -.  A. x  x  =  y
11 nfsb2 1998 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
129, 10, 11sbco2d 2027 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ y  /  x ] ph ) )
138, 12syl5rbbr 251 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
146, 13pm2.61i 156 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1527   [wsb 1629
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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