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Theorem sbco3 2137
Description: A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.)
Assertion
Ref Expression
sbco3  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )

Proof of Theorem sbco3
StepHypRef Expression
1 drsb1 2091 . . 3  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  y ] [
y  /  x ] ph ) )
2 nfa1 1845 . . . 4  |-  F/ x A. x  x  =  y
3 sbequ12a 1963 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
43sps 1814 . . . 4  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
52, 4sbbid 2118 . . 3  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
61, 5bitr3d 255 . 2  |-  ( A. x  x  =  y  ->  ( [ z  / 
y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
7 sbco 2129 . . . 4  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  [ x  /  y ] ph )
87sbbii 1718 . . 3  |-  ( [ z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
9 nfnae 2031 . . . 4  |-  F/ y  -.  A. x  x  =  y
10 nfna1 1851 . . . 4  |-  F/ x  -.  A. x  x  =  y
11 nfsb2 2073 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
129, 10, 11sbco2d 2136 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ y  /  x ] ph ) )
138, 12syl5rbbr 260 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
146, 13pm2.61i 164 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1377   [wsb 1711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712
This theorem is referenced by:  sbcom  2139
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