In a recent lecture the professor stated that given two complexity classes A and B, and given the existance of an oracle O such that $$A^o=B^o$$ (As I understand, meaning that a problem in A with can be reduced to a problem in B with the oracle O), It can be shown that classes A and B can not be seperated using a diagonilization argument.
Unfortunately he did not explain much further... Can someone give the outline\thought process of the general proof (informally)? I think I understand why diagonalization fails in specific time hirearchy proof constructions (due to the existence of a cook reduction) but I can not see how to generlize this in a satisfactory way.
Thank you.