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Theorem sbco2 2254
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 2093 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
2 sbequ 2215 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
31, 2bitr3d 263 . . 3  |-  ( z  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
43sps 1953 . 2  |-  ( A. z  z  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
5 nfnae 2162 . . 3  |-  F/ z  -.  A. z  z  =  y
6 sbco2.1 . . . 4  |-  F/ z
ph
76nfsb4 2229 . . 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 2248 . 2  |-  ( -. 
A. z  z  =  y  ->  ( [
y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
104, 9pm2.61i 169 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1452   F/wnf 1677   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  sbco2d  2255  equsb3ALT  2272  elsb3  2273  elsb4  2274  sb7f  2292  sbco4lem  2304  sbco4  2305  eqsb3  2566  clelsb3  2567  cbvab  2584  sbralie  3043  sbcco  3301  clelsb3f  28164  bj-clelsb3  31501
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