  We study 3-dimensional Poincar\'e duality pro-$p$ groups in the spirit of the
work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$
group $G$ has a nontrivial finitely presented subnormal subgroup of infinite
index, then either the subgroup is cyclic and normal, or the subgroup is cyclic
and the group is polycyclic, or the subgroup is Demushkin and normal in an open
subgroup of $G$. Also, we describe the centralizers of finitely generated
subgroups of 3-dimensional Poincar\'e duality pro-$p$ groups.
