Step |
Hyp |
Ref |
Expression |
1 |
|
ax-hvcom |
|- ( ( B e. ~H /\ C e. ~H ) -> ( B +h C ) = ( C +h B ) ) |
2 |
1
|
3adant1 |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( B +h C ) = ( C +h B ) ) |
3 |
2
|
oveq2d |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A -h ( B +h C ) ) = ( A -h ( C +h B ) ) ) |
4 |
|
hvsubass |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( A -h ( B +h C ) ) ) |
5 |
|
hvsubass |
|- ( ( A e. ~H /\ C e. ~H /\ B e. ~H ) -> ( ( A -h C ) -h B ) = ( A -h ( C +h B ) ) ) |
6 |
5
|
3com23 |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h C ) -h B ) = ( A -h ( C +h B ) ) ) |
7 |
3 4 6
|
3eqtr4d |
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( ( A -h C ) -h B ) ) |