Translate into a first order logic: "The equivalence classes with respect to $R$ of $a$ and $b$ are identical"

Question

Let $R$ be an equivalence relation on $X$Translate into a first-order logic: "The equivalence classes with respect to $R$ of $a$ and $b$ are identical"

I have given it some thought and have come to a conclusion - if two elements have identical classes of equivalence, then - whenever one of them is in a relation with another object, the second one must be in that relation, too. And so my solution is this:
$$(\forall x\in X)(aRx \iff bRx)$$ Is it the correct solution, or am I missing something?

Answer 1

I see two possibilities here:

1. If you just want the first order logic translation of the English sentence, then it's exactly what you wrote.
2. If you think that an equivalent, shorter version is acceptable (it might even be preferable), then, by noticing that two elements have the same equivalence class iff they are equivalent, you may consider to translate it as $a R b$.