Step |
Hyp |
Ref |
Expression |
1 |
|
eqrelf.1 |
|- F/_ x A |
2 |
|
eqrelf.2 |
|- F/_ x B |
3 |
|
eqrelf.3 |
|- F/_ y A |
4 |
|
eqrelf.4 |
|- F/_ y B |
5 |
|
eqrel |
|- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. u A. v ( <. u , v >. e. A <-> <. u , v >. e. B ) ) ) |
6 |
|
nfv |
|- F/ u ( <. x , y >. e. A <-> <. x , y >. e. B ) |
7 |
|
nfv |
|- F/ v ( <. x , y >. e. A <-> <. x , y >. e. B ) |
8 |
1
|
nfel2 |
|- F/ x <. u , v >. e. A |
9 |
2
|
nfel2 |
|- F/ x <. u , v >. e. B |
10 |
8 9
|
nfbi |
|- F/ x ( <. u , v >. e. A <-> <. u , v >. e. B ) |
11 |
3
|
nfel2 |
|- F/ y <. u , v >. e. A |
12 |
4
|
nfel2 |
|- F/ y <. u , v >. e. B |
13 |
11 12
|
nfbi |
|- F/ y ( <. u , v >. e. A <-> <. u , v >. e. B ) |
14 |
|
opeq12 |
|- ( ( x = u /\ y = v ) -> <. x , y >. = <. u , v >. ) |
15 |
14
|
eleq1d |
|- ( ( x = u /\ y = v ) -> ( <. x , y >. e. A <-> <. u , v >. e. A ) ) |
16 |
14
|
eleq1d |
|- ( ( x = u /\ y = v ) -> ( <. x , y >. e. B <-> <. u , v >. e. B ) ) |
17 |
15 16
|
bibi12d |
|- ( ( x = u /\ y = v ) -> ( ( <. x , y >. e. A <-> <. x , y >. e. B ) <-> ( <. u , v >. e. A <-> <. u , v >. e. B ) ) ) |
18 |
6 7 10 13 17
|
cbval2v |
|- ( A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) <-> A. u A. v ( <. u , v >. e. A <-> <. u , v >. e. B ) ) |
19 |
5 18
|
bitr4di |
|- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) ) |