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Theorem sbco2 2159
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 1993 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
2 sbequ 2118 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
31, 2bitr3d 255 . . 3  |-  ( z  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
43sps 1866 . 2  |-  ( A. z  z  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
5 nfnae 2059 . . 3  |-  F/ z  -.  A. z  z  =  y
6 sbco2.1 . . . 4  |-  F/ z
ph
76nfsb4 2132 . . 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 2152 . 2  |-  ( -. 
A. z  z  =  y  ->  ( [
y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
104, 9pm2.61i 164 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1393   F/wnf 1617   [wsb 1740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741
This theorem is referenced by:  sbco2d  2160  equsb3ALT  2178  elsb3  2179  elsb4  2180  sb7f  2198  sbco4lem  2210  sbco4  2211  2eu6OLD  2384  eqsb3  2577  clelsb3  2578  cbvab  2598  sbralie  3097  sbcco  3350  clelsb3f  27506  bj-clelsb3  34546  frege72  38085
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