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Theorem sbco2 2119
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 2117 . . . . 5  |-  ( [ x  /  z ] [ z  /  x ] ph  <->  ph )
3 sbequ 2093 . . . . 5  |-  ( x  =  y  ->  ( [ x  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
42, 3syl5bbr 251 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
5 sbequ12 1933 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5bitr3d 247 . . 3  |-  ( x  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
76sps 1762 . 2  |-  ( A. x  x  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
8 nfnae 2000 . . . 4  |-  F/ x  -.  A. x  x  =  y
91nfs1 2077 . . . . 5  |-  F/ x [ z  /  x ] ph
109nfsb4 2114 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  z ] [ z  /  x ] ph )
114a1i 11 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  <->  [ y  /  z ] [ z  /  x ] ph ) ) )
128, 10, 11sbied 2069 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
1312bicomd 193 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
147, 13pm2.61i 158 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1546   F/wnf 1550   [wsb 1655
This theorem is referenced by:  sbco2d  2120  equsb3  2135  elsb3  2136  elsb4  2137  sb7f  2153  2eu6  2323  eqsb3  2488  clelsb3  2489  sbralie  2888  sbcco  3126  clelsb3f  23815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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