  Let $R$ denote a commutative Noetherian ring. Brodmann et al. defined and
studied the concept of the local-global principle for annihilation of local
cohomology modules at level $r\in\mathbb{N}$ for the ideals $\frak a$ and
$\frak b$ of $R$. It was shown that this principle holds at levels 1,2, over
$R$ and at all levels whenever $\dim R\leq 4$. The goal of this paper is to
show that, if the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite or
$f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, then the local-global principle holds at all
levels $r\in\mathbb{N}_0$, for all ideals $\fa, \fb$ of $R$ and each finitely
generated $R$-module $M$, where $c_{\fa}^{\fb}(M)$ denotes the first non
$\fb$-cofiniteness of local cohomology module $H^i_{\fa}(M)$. As a consequence
of this, we provide a new and short proof of the Faltings' local-global
principle for finiteness dimensions. Also, several new results concerning the
finiteness dimensions are given.
