Assume for contradiction that B≠C 

Then either B⊈C or C⊈B 

We only need to disprove one of these since after we have 
disproved one of them, we can disprove the other just by 
swapping the roles of B and C.

We will assume B⊈C

Then ∃x such that x∈B and x∉C

The either x∈A or x∉A

Case 1: x∈A.  Then since x∈B, x∈A⋂B. But
since x∉C, x∉A⋂C. Therefore A⋂B≠A⋂C, a
contadiction since A⋂B=A⋂C is given. So case 1 is disproved.

Case 2: x∉A. Then since x∈B, x∈A⋃B. But
since x∉C, x∉A⋃C. Therefore A⋃B≠A⋃C, a
contadiction, since A⋃B=A⋃C is given. So case 2 is disproved.
   
Therefore B⊈C is false and B⊆C is true.

By swapping the roles of B and C in the above, C⊈B is false 
and C⊆B is true.

Therefore B=C.

Edwin