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Theorem wl-sbcom3 29612
Description: Substituting  y for  x and then  z for  y is equivalent to substituting  z for both  x and  y. Copy of ~? sbcom3OLD with a shortened proof.

Keep this theorem for a while here because an external reference to it exists.

(Contributed by Giovanni Mascellani, 8-Apr-2018.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
wl-sbcom3  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ z  /  y ] ph )

Proof of Theorem wl-sbcom3
StepHypRef Expression
1 nfa1 1845 . . . 4  |-  F/ y A. y  y  =  z
2 sbequ 2090 . . . . 5  |-  ( y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
32sps 1814 . . . 4  |-  ( A. y  y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph )
)
41, 3sbbid 2118 . . 3  |-  ( A. y  y  =  z  ->  ( [ z  / 
y ] [ y  /  x ] ph  <->  [ z  /  y ] [ z  /  x ] ph ) )
52pm5.74i 245 . . . . . 6  |-  ( ( y  =  z  ->  [ y  /  x ] ph )  <->  ( y  =  z  ->  [ z  /  x ] ph ) )
65albii 1620 . . . . 5  |-  ( A. y ( y  =  z  ->  [ y  /  x ] ph )  <->  A. y ( y  =  z  ->  [ z  /  x ] ph )
)
76a1i 11 . . . 4  |-  ( -. 
A. y  y  =  z  ->  ( A. y ( y  =  z  ->  [ y  /  x ] ph )  <->  A. y ( y  =  z  ->  [ z  /  x ] ph )
) )
8 sb4b 2071 . . . 4  |-  ( -. 
A. y  y  =  z  ->  ( [
z  /  y ] [ y  /  x ] ph  <->  A. y ( y  =  z  ->  [ y  /  x ] ph ) ) )
9 sb4b 2071 . . . 4  |-  ( -. 
A. y  y  =  z  ->  ( [
z  /  y ] [ z  /  x ] ph  <->  A. y ( y  =  z  ->  [ z  /  x ] ph ) ) )
107, 8, 93bitr4d 285 . . 3  |-  ( -. 
A. y  y  =  z  ->  ( [
z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ z  /  x ] ph ) )
114, 10pm2.61i 164 . 2  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ z  /  x ] ph )
12 sbcom 2139 . 2  |-  ( [ z  /  y ] [ z  /  x ] ph  <->  [ z  /  x ] [ z  /  y ] ph )
1311, 12bitri 249 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ z  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1377   [wsb 1711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712
This theorem is referenced by: (None)
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