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Theorem sbco2 2220
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.)
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 sbequ12 2057 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
2 sbequ 2181 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
31, 2bitr3d 258 . . 3  |-  ( z  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
43sps 1920 . 2  |-  ( A. z  z  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
5 nfnae 2124 . . 3  |-  F/ z  -.  A. z  z  =  y
6 sbco2.1 . . . 4  |-  F/ z
ph
76nfsb4 2195 . . 3  |-  ( -. 
A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
82a1i 11 . . 3  |-  ( -. 
A. z  z  =  y  ->  ( z  =  y  ->  ( [ z  /  x ] ph 
<->  [ y  /  x ] ph ) ) )
95, 7, 8sbied 2214 . 2  |-  ( -. 
A. z  z  =  y  ->  ( [
y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
104, 9pm2.61i 167 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187   A.wal 1435   F/wnf 1661   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by:  sbco2d  2221  equsb3ALT  2239  elsb3  2240  elsb4  2241  sb7f  2259  sbco4lem  2271  sbco4  2272  eqsb3  2533  clelsb3  2534  cbvab  2551  sbralie  3009  sbcco  3265  clelsb3f  28058  bj-clelsb3  31368
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