Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
|- x e. _V |
2 |
|
vex |
|- y e. _V |
3 |
1 2
|
opnzi |
|- <. x , y >. =/= (/) |
4 |
|
simpl |
|- ( ( (/) = <. x , y >. /\ ph ) -> (/) = <. x , y >. ) |
5 |
4
|
eqcomd |
|- ( ( (/) = <. x , y >. /\ ph ) -> <. x , y >. = (/) ) |
6 |
5
|
necon3ai |
|- ( <. x , y >. =/= (/) -> -. ( (/) = <. x , y >. /\ ph ) ) |
7 |
3 6
|
ax-mp |
|- -. ( (/) = <. x , y >. /\ ph ) |
8 |
7
|
nex |
|- -. E. y ( (/) = <. x , y >. /\ ph ) |
9 |
8
|
nex |
|- -. E. x E. y ( (/) = <. x , y >. /\ ph ) |
10 |
|
elopab |
|- ( (/) e. { <. x , y >. | ph } <-> E. x E. y ( (/) = <. x , y >. /\ ph ) ) |
11 |
9 10
|
mtbir |
|- -. (/) e. { <. x , y >. | ph } |
12 |
|
eleq1 |
|- ( <. A , B >. = (/) -> ( <. A , B >. e. { <. x , y >. | ph } <-> (/) e. { <. x , y >. | ph } ) ) |
13 |
11 12
|
mtbiri |
|- ( <. A , B >. = (/) -> -. <. A , B >. e. { <. x , y >. | ph } ) |
14 |
13
|
necon2ai |
|- ( <. A , B >. e. { <. x , y >. | ph } -> <. A , B >. =/= (/) ) |
15 |
|
opnz |
|- ( <. A , B >. =/= (/) <-> ( A e. _V /\ B e. _V ) ) |
16 |
14 15
|
sylib |
|- ( <. A , B >. e. { <. x , y >. | ph } -> ( A e. _V /\ B e. _V ) ) |
17 |
|
sbcex |
|- ( [. A / x ]. [. B / y ]. ph -> A e. _V ) |
18 |
|
spesbc |
|- ( [. A / x ]. [. B / y ]. ph -> E. x [. B / y ]. ph ) |
19 |
|
sbcex |
|- ( [. B / y ]. ph -> B e. _V ) |
20 |
19
|
exlimiv |
|- ( E. x [. B / y ]. ph -> B e. _V ) |
21 |
18 20
|
syl |
|- ( [. A / x ]. [. B / y ]. ph -> B e. _V ) |
22 |
17 21
|
jca |
|- ( [. A / x ]. [. B / y ]. ph -> ( A e. _V /\ B e. _V ) ) |
23 |
|
opeq1 |
|- ( z = A -> <. z , w >. = <. A , w >. ) |
24 |
23
|
eleq1d |
|- ( z = A -> ( <. z , w >. e. { <. x , y >. | ph } <-> <. A , w >. e. { <. x , y >. | ph } ) ) |
25 |
|
dfsbcq2 |
|- ( z = A -> ( [ z / x ] [ w / y ] ph <-> [. A / x ]. [ w / y ] ph ) ) |
26 |
24 25
|
bibi12d |
|- ( z = A -> ( ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph ) <-> ( <. A , w >. e. { <. x , y >. | ph } <-> [. A / x ]. [ w / y ] ph ) ) ) |
27 |
|
opeq2 |
|- ( w = B -> <. A , w >. = <. A , B >. ) |
28 |
27
|
eleq1d |
|- ( w = B -> ( <. A , w >. e. { <. x , y >. | ph } <-> <. A , B >. e. { <. x , y >. | ph } ) ) |
29 |
|
dfsbcq2 |
|- ( w = B -> ( [ w / y ] ph <-> [. B / y ]. ph ) ) |
30 |
29
|
sbcbidv |
|- ( w = B -> ( [. A / x ]. [ w / y ] ph <-> [. A / x ]. [. B / y ]. ph ) ) |
31 |
28 30
|
bibi12d |
|- ( w = B -> ( ( <. A , w >. e. { <. x , y >. | ph } <-> [. A / x ]. [ w / y ] ph ) <-> ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph ) ) ) |
32 |
|
vopelopabsb |
|- ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph ) |
33 |
26 31 32
|
vtocl2g |
|- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph ) ) |
34 |
16 22 33
|
pm5.21nii |
|- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph ) |