Step |
Hyp |
Ref |
Expression |
1 |
|
copsex2d.xph |
|- ( ph -> A. x ph ) |
2 |
|
copsex2d.yph |
|- ( ph -> A. y ph ) |
3 |
|
copsex2d.xch |
|- ( ph -> F/ x ch ) |
4 |
|
copsex2d.ych |
|- ( ph -> F/ y ch ) |
5 |
|
copsex2d.exa |
|- ( ph -> A e. U ) |
6 |
|
copsex2d.exb |
|- ( ph -> B e. V ) |
7 |
|
copsex2d.is |
|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) ) |
8 |
|
elisset |
|- ( A e. U -> E. x x = A ) |
9 |
5 8
|
syl |
|- ( ph -> E. x x = A ) |
10 |
|
elisset |
|- ( B e. V -> E. y y = B ) |
11 |
6 10
|
syl |
|- ( ph -> E. y y = B ) |
12 |
|
exdistrv |
|- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) ) |
13 |
|
nfe1 |
|- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) |
14 |
13
|
a1i |
|- ( ph -> F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) ) |
15 |
14 3
|
nfbid |
|- ( ph -> F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) |
16 |
15
|
19.9d |
|- ( ph -> ( E. x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) ) |
17 |
|
nfe1 |
|- F/ y E. y ( <. A , B >. = <. x , y >. /\ ps ) |
18 |
17
|
a1i |
|- ( ph -> F/ y E. y ( <. A , B >. = <. x , y >. /\ ps ) ) |
19 |
1 18
|
bj-nfexd |
|- ( ph -> F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) ) |
20 |
19 4
|
nfbid |
|- ( ph -> F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) |
21 |
20
|
19.9d |
|- ( ph -> ( E. y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) ) |
22 |
|
opeq12 |
|- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. ) |
23 |
|
copsexgw |
|- ( <. A , B >. = <. x , y >. -> ( ps <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) ) ) |
24 |
23
|
bicomd |
|- ( <. A , B >. = <. x , y >. -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ps ) ) |
25 |
24
|
eqcoms |
|- ( <. x , y >. = <. A , B >. -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ps ) ) |
26 |
22 25
|
syl |
|- ( ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ps ) ) |
27 |
26
|
adantl |
|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ps ) ) |
28 |
27 7
|
bitrd |
|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) |
29 |
28
|
ex |
|- ( ph -> ( ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) ) |
30 |
2 21 29
|
bj-exlimd |
|- ( ph -> ( E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) ) |
31 |
1 16 30
|
bj-exlimd |
|- ( ph -> ( E. x E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) ) |
32 |
12 31
|
syl5bir |
|- ( ph -> ( ( E. x x = A /\ E. y y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) ) |
33 |
9 11 32
|
mp2and |
|- ( ph -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) |