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Theorem sbco2 1980
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sbco2.1  |-  F/ z
ph
Assertion
Ref Expression
sbco2  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )

Proof of Theorem sbco2
StepHypRef Expression
1 sbco2.1 . . . . . 6  |-  F/ z
ph
21sbid2 1978 . . . . 5  |-  ( [ x  /  z ] [ z  /  x ] ph  <->  ph )
3 sbequ 1952 . . . . 5  |-  ( x  =  y  ->  ( [ x  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
42, 3syl5bbr 252 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
5 sbequ12 1892 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5bitr3d 248 . . 3  |-  ( x  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
76a4s 1700 . 2  |-  ( A. x  x  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
8 nfnae 1846 . . . 4  |-  F/ x  -.  A. x  x  =  y
91nfs1 1921 . . . . 5  |-  F/ x [ z  /  x ] ph
109nfsb4 1973 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  z ] [ z  /  x ] ph )
114a1i 12 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  <->  [ y  /  z ] [ z  /  x ] ph ) ) )
128, 10, 11sbied 1908 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
1312bicomd 194 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
147, 13pm2.61i 158 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178   A.wal 1532   F/wnf 1539    = wceq 1619   [wsb 1882
This theorem is referenced by:  sbco2d  1981  equsb3  2062  elsb3  2063  elsb4  2064  dfsb7  2079  sb7f  2080  2eu6  2198  eqsb3  2350  clelsb3  2351  sbralie  2716  sbcco  2943
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883
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