Double Quotient by Ideals in Local Rings

Question

Suppose I have a local morphism of local noetherian rings $\phi:A\to B$, $m_A, m_B$ the maximal ideals. Then the $m_B$-adic completion of $B/m_AB$ is: $$(B/m_AB)\hat{}= \varprojlim(B/m_AB/m_B^n) $$ I wanted to know if there is a better way to write that "double quotient" better, as a single quotient, in particular relating that to $m_A^nB$. My guess by now is that $(B/m_AB)/m_B^n=B/(m_AB,m^n_B)$, but still I wanted to find a better relation.