Constraints on each node and line in power systems generally have upper and lower bounds, denoted as two-sided constraints. Most existing power system optimization methods with the distributionally robust (DR) chance-constrained program treat the two-sided DR chance constraint separately, which is an inexact approximation. This letter derives an equivalent reformulation for the generic two-sided DR chance constraint under the interval moment based ambiguity set, which does not require the exact moment information. The derived reformulation is a second-order cone program (SOCP) formulation and is then applied to the optimal power flow (OPF) problem under uncertainty. Numerical results on several IEEE systems demonstrate the effectiveness of the proposed SOCP formulation and show the differences with other DR chance-constrained OPF approaches.