Step |
Hyp |
Ref |
Expression |
1 |
|
cnveq |
|- ( A = B -> `' A = `' B ) |
2 |
|
dfrel2 |
|- ( Rel A <-> `' `' A = A ) |
3 |
|
dfrel2 |
|- ( Rel B <-> `' `' B = B ) |
4 |
|
cnveq |
|- ( `' A = `' B -> `' `' A = `' `' B ) |
5 |
|
eqeq2 |
|- ( B = `' `' B -> ( `' `' A = B <-> `' `' A = `' `' B ) ) |
6 |
4 5
|
syl5ibr |
|- ( B = `' `' B -> ( `' A = `' B -> `' `' A = B ) ) |
7 |
6
|
eqcoms |
|- ( `' `' B = B -> ( `' A = `' B -> `' `' A = B ) ) |
8 |
3 7
|
sylbi |
|- ( Rel B -> ( `' A = `' B -> `' `' A = B ) ) |
9 |
|
eqeq1 |
|- ( A = `' `' A -> ( A = B <-> `' `' A = B ) ) |
10 |
9
|
imbi2d |
|- ( A = `' `' A -> ( ( `' A = `' B -> A = B ) <-> ( `' A = `' B -> `' `' A = B ) ) ) |
11 |
8 10
|
syl5ibr |
|- ( A = `' `' A -> ( Rel B -> ( `' A = `' B -> A = B ) ) ) |
12 |
11
|
eqcoms |
|- ( `' `' A = A -> ( Rel B -> ( `' A = `' B -> A = B ) ) ) |
13 |
2 12
|
sylbi |
|- ( Rel A -> ( Rel B -> ( `' A = `' B -> A = B ) ) ) |
14 |
13
|
imp |
|- ( ( Rel A /\ Rel B ) -> ( `' A = `' B -> A = B ) ) |
15 |
1 14
|
impbid2 |
|- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) ) |