Use of the expression "for some element $x$ $\in$ a subgroup $H$"

Question

This is a question asking for clarification in use of the terms such as "for some element..." and "let $x$ be an arbitrary element in a set...." in abstract algebra. Let's say I have a group $M$ and it has two subgroups $A$ and $B$. Now also, let's say that $m_1$ and $m_2$ are two elements in $M$ and they are related by a relation $R$ iff $am_1b$ = $m_2$ for some element $a$ $\in$ $A$ and $b$ $\in$ $B$. So, my confusion arose in proving symmetry. I have $am_1b$ = $m_2$. So does it mean, I have to show that $am_2b$ = $m_1$ for the same $a$ and $b$? Cause what I am thinking is that just by applying inverses of $a$ and $b$ , I will get $m_1$ = $a^{-1}m_2b^{-1}$, and since $A$ and $B$ are subgroups of $M$, the inverses of $a$ and $b$ will exist in $A$ and $B$ respectively. So, is the symmetry property verified?