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Theorem wl-sbcom2d 28385
Description: Version of sbcom2 2151 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
Hypotheses
Ref Expression
wl-sbcom2d.1  |-  ( ph  ->  -.  A. x  x  =  w )
wl-sbcom2d.2  |-  ( ph  ->  -.  A. x  x  =  z )
wl-sbcom2d.3  |-  ( ph  ->  -.  A. z  z  =  y )
Assertion
Ref Expression
wl-sbcom2d  |-  ( ph  ->  ( [ w  / 
z ] [ y  /  x ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) )

Proof of Theorem wl-sbcom2d
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax6ev 1710 . 2  |-  E. u  u  =  y
2 ax6ev 1710 . 2  |-  E. v 
v  =  w
3 wl-sbcom2d.2 . . . . . . . . 9  |-  ( ph  ->  -.  A. x  x  =  z )
4 wl-sbcom2d-lem2 28384 . . . . . . . . . . . 12  |-  ( -. 
A. z  z  =  x  ->  ( [
u  /  x ] [ v  /  z ] ps  <->  A. x A. z
( ( x  =  u  /\  z  =  v )  ->  ps ) ) )
5 alcom 1783 . . . . . . . . . . . . 13  |-  ( A. x A. z ( ( x  =  u  /\  z  =  v )  ->  ps )  <->  A. z A. x ( ( x  =  u  /\  z  =  v )  ->  ps ) )
6 ancom 450 . . . . . . . . . . . . . . 15  |-  ( ( x  =  u  /\  z  =  v )  <->  ( z  =  v  /\  x  =  u )
)
76imbi1i 325 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  u  /\  z  =  v )  ->  ps )  <->  ( ( z  =  v  /\  x  =  u )  ->  ps )
)
872albii 1611 . . . . . . . . . . . . 13  |-  ( A. z A. x ( ( x  =  u  /\  z  =  v )  ->  ps )  <->  A. z A. x ( ( z  =  v  /\  x  =  u )  ->  ps ) )
95, 8bitri 249 . . . . . . . . . . . 12  |-  ( A. x A. z ( ( x  =  u  /\  z  =  v )  ->  ps )  <->  A. z A. x ( ( z  =  v  /\  x  =  u )  ->  ps ) )
104, 9syl6bb 261 . . . . . . . . . . 11  |-  ( -. 
A. z  z  =  x  ->  ( [
u  /  x ] [ v  /  z ] ps  <->  A. z A. x
( ( z  =  v  /\  x  =  u )  ->  ps ) ) )
1110naecoms 2000 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  z  ->  ( [
u  /  x ] [ v  /  z ] ps  <->  A. z A. x
( ( z  =  v  /\  x  =  u )  ->  ps ) ) )
12 wl-sbcom2d-lem2 28384 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  z  ->  ( [
v  /  z ] [ u  /  x ] ps  <->  A. z A. x
( ( z  =  v  /\  x  =  u )  ->  ps ) ) )
1311, 12bitr4d 256 . . . . . . . . 9  |-  ( -. 
A. x  x  =  z  ->  ( [
u  /  x ] [ v  /  z ] ps  <->  [ v  /  z ] [ u  /  x ] ps ) )
143, 13syl 16 . . . . . . . 8  |-  ( ph  ->  ( [ u  /  x ] [ v  / 
z ] ps  <->  [ v  /  z ] [
u  /  x ] ps ) )
1514adantl 466 . . . . . . 7  |-  ( ( ( u  =  y  /\  v  =  w )  /\  ph )  ->  ( [ u  /  x ] [ v  / 
z ] ps  <->  [ v  /  z ] [
u  /  x ] ps ) )
16 wl-sbcom2d.3 . . . . . . . . . 10  |-  ( ph  ->  -.  A. z  z  =  y )
17 wl-sbcom2d-lem1 28383 . . . . . . . . . 10  |-  ( ( v  =  w  /\  u  =  y )  ->  ( -.  A. z 
z  =  y  -> 
( [ v  / 
z ] [ u  /  x ] ps  <->  [ w  /  z ] [
y  /  x ] ps ) ) )
1816, 17syl5 32 . . . . . . . . 9  |-  ( ( v  =  w  /\  u  =  y )  ->  ( ph  ->  ( [ v  /  z ] [ u  /  x ] ps  <->  [ w  /  z ] [ y  /  x ] ps ) ) )
1918ancoms 453 . . . . . . . 8  |-  ( ( u  =  y  /\  v  =  w )  ->  ( ph  ->  ( [ v  /  z ] [ u  /  x ] ps  <->  [ w  /  z ] [ y  /  x ] ps ) ) )
2019imp 429 . . . . . . 7  |-  ( ( ( u  =  y  /\  v  =  w )  /\  ph )  ->  ( [ v  / 
z ] [ u  /  x ] ps  <->  [ w  /  z ] [
y  /  x ] ps ) )
2115, 20bitrd 253 . . . . . 6  |-  ( ( ( u  =  y  /\  v  =  w )  /\  ph )  ->  ( [ u  /  x ] [ v  / 
z ] ps  <->  [ w  /  z ] [
y  /  x ] ps ) )
22 wl-sbcom2d.1 . . . . . . . 8  |-  ( ph  ->  -.  A. x  x  =  w )
23 wl-sbcom2d-lem1 28383 . . . . . . . 8  |-  ( ( u  =  y  /\  v  =  w )  ->  ( -.  A. x  x  =  w  ->  ( [ u  /  x ] [ v  /  z ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) ) )
2422, 23syl5 32 . . . . . . 7  |-  ( ( u  =  y  /\  v  =  w )  ->  ( ph  ->  ( [ u  /  x ] [ v  /  z ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) ) )
2524imp 429 . . . . . 6  |-  ( ( ( u  =  y  /\  v  =  w )  /\  ph )  ->  ( [ u  /  x ] [ v  / 
z ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) )
2621, 25bitr3d 255 . . . . 5  |-  ( ( ( u  =  y  /\  v  =  w )  /\  ph )  ->  ( [ w  / 
z ] [ y  /  x ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) )
2726exp31 604 . . . 4  |-  ( u  =  y  ->  (
v  =  w  -> 
( ph  ->  ( [ w  /  z ] [ y  /  x ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) ) ) )
2827exlimdv 1690 . . 3  |-  ( u  =  y  ->  ( E. v  v  =  w  ->  ( ph  ->  ( [ w  /  z ] [ y  /  x ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) ) ) )
2928exlimiv 1688 . 2  |-  ( E. u  u  =  y  ->  ( E. v 
v  =  w  -> 
( ph  ->  ( [ w  /  z ] [ y  /  x ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) ) ) )
301, 2, 29mp2 9 1  |-  ( ph  ->  ( [ w  / 
z ] [ y  /  x ] ps  <->  [ y  /  x ] [ w  /  z ] ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367   E.wex 1586   [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)
  Copyright terms: Public domain W3C validator