Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-sbcom3 Structured version   Unicode version

Theorem wl-sbcom3 30197
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 1898 . . . 4  |-  F/ y A. y  y  =  z
2 sbequ 2118 . . . . 5  |-  ( y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
32sps 1866 . . . 4  |-  ( A. y  y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph )
)
41, 3sbbid 2145 . . 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 1641 . . . . 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 2099 . . . 4  |-  ( -. 
A. y  y  =  z  ->  ( [
z  /  y ] [ y  /  x ] ph  <->  A. y ( y  =  z  ->  [ y  /  x ] ph ) ) )
9 sb4b 2099 . . . 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 2162 . 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 1393   [wsb 1740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator