Step |
Hyp |
Ref |
Expression |
1 |
|
addcom |
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) |
2 |
1
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - C ) = ( ( B + A ) - C ) ) |
3 |
2
|
3adant3 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( ( B + A ) - C ) ) |
4 |
|
addsubass |
|- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + A ) - C ) = ( B + ( A - C ) ) ) |
5 |
4
|
3com12 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) - C ) = ( B + ( A - C ) ) ) |
6 |
|
subcl |
|- ( ( A e. CC /\ C e. CC ) -> ( A - C ) e. CC ) |
7 |
|
addcom |
|- ( ( B e. CC /\ ( A - C ) e. CC ) -> ( B + ( A - C ) ) = ( ( A - C ) + B ) ) |
8 |
6 7
|
sylan2 |
|- ( ( B e. CC /\ ( A e. CC /\ C e. CC ) ) -> ( B + ( A - C ) ) = ( ( A - C ) + B ) ) |
9 |
8
|
3impb |
|- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( B + ( A - C ) ) = ( ( A - C ) + B ) ) |
10 |
9
|
3com12 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + ( A - C ) ) = ( ( A - C ) + B ) ) |
11 |
3 5 10
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( ( A - C ) + B ) ) |