  Bosonic topological insulators (BTI) in three dimensions are
symmetry-protected topological phases (SPT) protected by time-reversal and
boson number conservation {symmetries}. BTI in three dimensions were first
proposed and classified by the group cohomology theory which suggests two
distinct root states, each carrying a $\mathbb{Z}_2$ index. Soon after, surface
anomalous topological orders were proposed to identify different root states of
BTI, which even leads to a new BTI root state beyond the group cohomology
classification. In this paper, we propose a universal physical mechanism via
\textit{vortex-line condensation} {from} a 3d superfluid to achieve all {three}
root states. It naturally produces bulk topological quantum field theory (TQFT)
description for each root state. Topologically ordered states on the surface
are \textit{rigorously} derived by placing TQFT on an open manifold, which
allows us to explicitly demonstrate the bulk-boundary correspondence. Finally,
we generalize the mechanism to $Z_N$ symmetries and discuss potential SPT
phases beyond the group cohomology classification.
