Step |
Hyp |
Ref |
Expression |
1 |
|
log11d.a |
|- ( ph -> A e. CC ) |
2 |
|
log11d.b |
|- ( ph -> B e. CC ) |
3 |
|
log11d.1 |
|- ( ph -> A =/= 0 ) |
4 |
|
log11d.2 |
|- ( ph -> B =/= 0 ) |
5 |
|
fveq2 |
|- ( ( log ` A ) = ( log ` B ) -> ( exp ` ( log ` A ) ) = ( exp ` ( log ` B ) ) ) |
6 |
|
eflog |
|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = A ) |
7 |
1 3 6
|
syl2anc |
|- ( ph -> ( exp ` ( log ` A ) ) = A ) |
8 |
|
eflog |
|- ( ( B e. CC /\ B =/= 0 ) -> ( exp ` ( log ` B ) ) = B ) |
9 |
2 4 8
|
syl2anc |
|- ( ph -> ( exp ` ( log ` B ) ) = B ) |
10 |
7 9
|
eqeq12d |
|- ( ph -> ( ( exp ` ( log ` A ) ) = ( exp ` ( log ` B ) ) <-> A = B ) ) |
11 |
5 10
|
imbitrid |
|- ( ph -> ( ( log ` A ) = ( log ` B ) -> A = B ) ) |
12 |
|
fveq2 |
|- ( A = B -> ( log ` A ) = ( log ` B ) ) |
13 |
11 12
|
impbid1 |
|- ( ph -> ( ( log ` A ) = ( log ` B ) <-> A = B ) ) |