This paper contributes to the question of under which conditions recursively enumerable sets with isomorphic lattices of recursively enumerable supersets are automorphic in the lattice of all recursively enumerable sets. We show that hyperhypersimple sets (i.e. sets where the recursively enumerable supersets form a Boolean algebra) are automorphic if there is a $\Sigma^0_3$-definable isomorphism between their lattices of supersets. Lerman, Shore and Soare have shown that this is not true if one replaces $\Sigma^0_3$ by $\Sigma^0_4$.