Step |
Hyp |
Ref |
Expression |
1 |
|
goaleq12d.1 |
|- ( ph -> M = N ) |
2 |
|
goaleq12d.2 |
|- ( ph -> A = B ) |
3 |
|
df-goal |
|- A.g M A = <. 2o , <. M , A >. >. |
4 |
3
|
a1i |
|- ( ph -> A.g M A = <. 2o , <. M , A >. >. ) |
5 |
1 2
|
opeq12d |
|- ( ph -> <. M , A >. = <. N , B >. ) |
6 |
5
|
opeq2d |
|- ( ph -> <. 2o , <. M , A >. >. = <. 2o , <. N , B >. >. ) |
7 |
|
df-goal |
|- A.g N B = <. 2o , <. N , B >. >. |
8 |
7
|
eqcomi |
|- <. 2o , <. N , B >. >. = A.g N B |
9 |
8
|
a1i |
|- ( ph -> <. 2o , <. N , B >. >. = A.g N B ) |
10 |
6 9
|
eqtrd |
|- ( ph -> <. 2o , <. M , A >. >. = A.g N B ) |
11 |
4 10
|
eqtrd |
|- ( ph -> A.g M A = A.g N B ) |