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Theorem sbco2OLD 2112
Description: Obsolete proof of sbco2 2111 as of 17-Sep-2018. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
sbco2.1  |-  F/ z
ph
Assertion
Ref Expression
sbco2OLD  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )

Proof of Theorem sbco2OLD
StepHypRef Expression
1 sbco2.1 . . . . . 6  |-  F/ z
ph
21sbid2 2108 . . . . 5  |-  ( [ x  /  z ] [ z  /  x ] ph  <->  ph )
3 sbequ 2067 . . . . 5  |-  ( x  =  y  ->  ( [ x  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph ) )
42, 3syl5bbr 259 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
5 sbequ12 1936 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5bitr3d 255 . . 3  |-  ( x  =  y  ->  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
76sps 1800 . 2  |-  ( A. x  x  =  y  ->  ( [ y  / 
z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
8 nfnae 2005 . . . 4  |-  F/ x  -.  A. x  x  =  y
91nfs1 2054 . . . . 5  |-  F/ x [ z  /  x ] ph
109nfsb4 2081 . . . 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 2102 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<->  [ y  /  z ] [ z  /  x ] ph ) )
1312bicomd 201 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
147, 13pm2.61i 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: (None)
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