Let \(\textbf{K}\) and \(\textbf{M}\) be locally finite quasivarieties of finite type such that \(\textbf{K}\subset \textbf{M}\). If \(\textbf{K}\) is profinite then the filter \([\textbf{K},\textbf{M}]\) in the quasivariety lattice \(\textrm{Lq}(\textbf{M})\) is an atomic lattice and \(\textbf{K}\) has an independent quasi-equational basis relative to \(\textbf{M}\). Applications of these results for lattices, unary algebras, groups, unary algebras, and distributive algebras are presented which concern some well-known problems on standard topological quasivarieties and other problems.