Step |
Hyp |
Ref |
Expression |
1 |
|
wl-sbcom2d.1 |
|- ( ph -> -. A. x x = w ) |
2 |
|
wl-sbcom2d.2 |
|- ( ph -> -. A. x x = z ) |
3 |
|
wl-sbcom2d.3 |
|- ( ph -> -. A. z z = y ) |
4 |
|
ax6ev |
|- E. u u = y |
5 |
|
ax6ev |
|- E. v v = w |
6 |
|
wl-sbcom2d-lem2 |
|- ( -. A. z z = x -> ( [ u / x ] [ v / z ] ps <-> A. x A. z ( ( x = u /\ z = v ) -> ps ) ) ) |
7 |
|
alcom |
|- ( A. x A. z ( ( x = u /\ z = v ) -> ps ) <-> A. z A. x ( ( x = u /\ z = v ) -> ps ) ) |
8 |
|
ancomst |
|- ( ( ( x = u /\ z = v ) -> ps ) <-> ( ( z = v /\ x = u ) -> ps ) ) |
9 |
8
|
2albii |
|- ( A. z A. x ( ( x = u /\ z = v ) -> ps ) <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) |
10 |
7 9
|
bitri |
|- ( A. x A. z ( ( x = u /\ z = v ) -> ps ) <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) |
11 |
6 10
|
bitrdi |
|- ( -. A. z z = x -> ( [ u / x ] [ v / z ] ps <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) ) |
12 |
11
|
naecoms |
|- ( -. A. x x = z -> ( [ u / x ] [ v / z ] ps <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) ) |
13 |
|
wl-sbcom2d-lem2 |
|- ( -. A. x x = z -> ( [ v / z ] [ u / x ] ps <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) ) |
14 |
12 13
|
bitr4d |
|- ( -. A. x x = z -> ( [ u / x ] [ v / z ] ps <-> [ v / z ] [ u / x ] ps ) ) |
15 |
2 14
|
syl |
|- ( ph -> ( [ u / x ] [ v / z ] ps <-> [ v / z ] [ u / x ] ps ) ) |
16 |
15
|
adantl |
|- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ u / x ] [ v / z ] ps <-> [ v / z ] [ u / x ] ps ) ) |
17 |
|
wl-sbcom2d-lem1 |
|- ( ( u = y /\ v = w ) -> ( -. A. x x = w -> ( [ u / x ] [ v / z ] ps <-> [ y / x ] [ w / z ] ps ) ) ) |
18 |
1 17
|
syl5 |
|- ( ( u = y /\ v = w ) -> ( ph -> ( [ u / x ] [ v / z ] ps <-> [ y / x ] [ w / z ] ps ) ) ) |
19 |
18
|
imp |
|- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ u / x ] [ v / z ] ps <-> [ y / x ] [ w / z ] ps ) ) |
20 |
|
wl-sbcom2d-lem1 |
|- ( ( v = w /\ u = y ) -> ( -. A. z z = y -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) ) ) |
21 |
3 20
|
syl5 |
|- ( ( v = w /\ u = y ) -> ( ph -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) ) ) |
22 |
21
|
ancoms |
|- ( ( u = y /\ v = w ) -> ( ph -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) ) ) |
23 |
22
|
imp |
|- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) ) |
24 |
16 19 23
|
3bitr3rd |
|- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) |
25 |
24
|
exp31 |
|- ( u = y -> ( v = w -> ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) ) ) |
26 |
25
|
exlimdv |
|- ( u = y -> ( E. v v = w -> ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) ) ) |
27 |
26
|
exlimiv |
|- ( E. u u = y -> ( E. v v = w -> ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) ) ) |
28 |
4 5 27
|
mp2 |
|- ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) |