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Theorem sbco2 2111
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 1936 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
2 sbequ 2067 . . . 4  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
31, 2bitr3d 255 . . 3  |-  ( z  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
43sps 1800 . 2  |-  ( A. z  z  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
5 nfna1 1837 . . 3  |-  F/ z  -.  A. z  z  =  y
6 sbco2.1 . . . 4  |-  F/ z
ph
76nfsb4 2081 . . 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 2102 . 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 1367   F/wnf 1589   [wsb 1700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701
This theorem is referenced by:  sbco2d  2113  equsb3ALT  2138  elsb3  2139  elsb4  2140  sb7f  2163  sbco4lem  2178  sbco4  2179  2eu6OLD  2372  eqsb3  2544  clelsb3  2545  sbralie  2960  sbcco  3209  clelsb3f  25864  bj-clelsb3  32359
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