Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
|- a e. _V |
2 |
|
vex |
|- b e. _V |
3 |
1 2
|
pm3.2i |
|- ( a e. _V /\ b e. _V ) |
4 |
3
|
a1i |
|- ( ( A e. X /\ B e. X ) -> ( a e. _V /\ b e. _V ) ) |
5 |
4
|
anim1ci |
|- ( ( ( A e. X /\ B e. X ) /\ ph ) -> ( ph /\ ( a e. _V /\ b e. _V ) ) ) |
6 |
5
|
adantr |
|- ( ( ( ( A e. X /\ B e. X ) /\ ph ) /\ ( A = a /\ B = b ) ) -> ( ph /\ ( a e. _V /\ b e. _V ) ) ) |
7 |
|
sbcid |
|- ( [. b / b ]. [. a / a ]. ph <-> [. a / a ]. ph ) |
8 |
|
sbcid |
|- ( [. a / a ]. ph <-> ph ) |
9 |
7 8
|
sylbbr |
|- ( ph -> [. b / b ]. [. a / a ]. ph ) |
10 |
9
|
adantl |
|- ( ( ( A e. X /\ B e. X ) /\ ph ) -> [. b / b ]. [. a / a ]. ph ) |
11 |
|
opeq12 |
|- ( ( A = a /\ B = b ) -> <. A , B >. = <. a , b >. ) |
12 |
10 11
|
anim12ci |
|- ( ( ( ( A e. X /\ B e. X ) /\ ph ) /\ ( A = a /\ B = b ) ) -> ( <. A , B >. = <. a , b >. /\ [. b / b ]. [. a / a ]. ph ) ) |
13 |
|
nfv |
|- F/ x ( <. A , B >. = <. a , b >. /\ [. b / b ]. [. a / a ]. ph ) |
14 |
|
nfv |
|- F/ y ( <. A , B >. = <. a , b >. /\ [. b / b ]. [. a / a ]. ph ) |
15 |
|
opeq12 |
|- ( ( x = a /\ y = b ) -> <. x , y >. = <. a , b >. ) |
16 |
15
|
eqeq2d |
|- ( ( x = a /\ y = b ) -> ( <. A , B >. = <. x , y >. <-> <. A , B >. = <. a , b >. ) ) |
17 |
|
dfsbcq |
|- ( y = b -> ( [. y / b ]. [. x / a ]. ph <-> [. b / b ]. [. x / a ]. ph ) ) |
18 |
|
dfsbcq |
|- ( x = a -> ( [. x / a ]. ph <-> [. a / a ]. ph ) ) |
19 |
18
|
sbcbidv |
|- ( x = a -> ( [. b / b ]. [. x / a ]. ph <-> [. b / b ]. [. a / a ]. ph ) ) |
20 |
17 19
|
sylan9bbr |
|- ( ( x = a /\ y = b ) -> ( [. y / b ]. [. x / a ]. ph <-> [. b / b ]. [. a / a ]. ph ) ) |
21 |
16 20
|
anbi12d |
|- ( ( x = a /\ y = b ) -> ( ( <. A , B >. = <. x , y >. /\ [. y / b ]. [. x / a ]. ph ) <-> ( <. A , B >. = <. a , b >. /\ [. b / b ]. [. a / a ]. ph ) ) ) |
22 |
21
|
adantl |
|- ( ( ph /\ ( x = a /\ y = b ) ) -> ( ( <. A , B >. = <. x , y >. /\ [. y / b ]. [. x / a ]. ph ) <-> ( <. A , B >. = <. a , b >. /\ [. b / b ]. [. a / a ]. ph ) ) ) |
23 |
13 14 22
|
spc2ed |
|- ( ( ph /\ ( a e. _V /\ b e. _V ) ) -> ( ( <. A , B >. = <. a , b >. /\ [. b / b ]. [. a / a ]. ph ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ [. y / b ]. [. x / a ]. ph ) ) ) |
24 |
6 12 23
|
sylc |
|- ( ( ( ( A e. X /\ B e. X ) /\ ph ) /\ ( A = a /\ B = b ) ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ [. y / b ]. [. x / a ]. ph ) ) |
25 |
24
|
exp31 |
|- ( ( A e. X /\ B e. X ) -> ( ph -> ( ( A = a /\ B = b ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ [. y / b ]. [. x / a ]. ph ) ) ) ) |
26 |
25
|
com23 |
|- ( ( A e. X /\ B e. X ) -> ( ( A = a /\ B = b ) -> ( ph -> E. x E. y ( <. A , B >. = <. x , y >. /\ [. y / b ]. [. x / a ]. ph ) ) ) ) |