In this paper we give an analytic proof of the identity $A_{5,3,3}(n) =B^0_{5,3,3}(n)$, where $A_{5,3,3}(n)$ counts the number of partitions of $n$ subject to certain restrictions on their parts, and $B^0_{5,3,3}(n)$ counts the number of partitions of $n$ subject to certain other restrictions on their parts, both too long to be stated in the abstract. Our proof establishes actually a refinement of that partition identity. The original identity was first discovered by the first author jointly with M. Ruby Salestina and S. R. Sudarshan in ["A new theorem on partitions," Proc. Int. Conference on Special Functions, IMSC, Chennai, India, September 23-27, 2002; to appear], where it was also given a combinatorial proof, thus responding a question of Andrews.