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Theorem sbco2 2025
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 2023 . . . . 5  |-  ( [ x  /  z ] [ z  /  x ] ph  <->  ph )
3 sbequ 1999 . . . . 5  |-  ( x  =  y  ->  ( [ x  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
42, 3syl5bbr 250 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
5 sbequ12 1862 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5bitr3d 246 . . 3  |-  ( x  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
76sps 1741 . 2  |-  ( A. x  x  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
8 nfnae 1899 . . . 4  |-  F/ x  -.  A. x  x  =  y
91nfs1 1989 . . . . 5  |-  F/ x [ z  /  x ] ph
109nfsb4 2020 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  z ] [ z  /  x ] ph )
114a1i 10 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  <->  [ y  /  z ] [ z  /  x ] ph ) ) )
128, 10, 11sbied 1981 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
1312bicomd 192 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
147, 13pm2.61i 156 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531   [wsb 1630
This theorem is referenced by:  sbco2d  2026  equsb3  2041  elsb3  2042  elsb4  2043  dfsb7  2061  sb7f  2062  2eu6  2229  eqsb3  2385  clelsb3  2386  sbralie  2778  sbcco  3014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631
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