Existential Elimination :
Now we have a sentence, A, is with an existentially quantified variable, v, so then just for every constant symbol k, that it does not appear anywhere else in the knowledge base than we can substitute k for v in A as:
∃ v A /Subst({v/k}, A)
Now we have an example if we know that ∃X (likes(X,ice_cream)), so then we can choose a particular name for X. Thus we could choose ben for this, giving us: likes(ben, ice_cream), if only the constant ben does not appear somewhere else in our knowledge base.
Because of that, why the condition about the existential variable being unique to the new sentence? However basically, what you are doing here is giving a particular name to a variable you know must exist. Means it would be unwise to give this a name that already exists. Now we have or example, assume that we have the predicates brother(john,X), sister(john, susan) so then, where instantiating X, there it would be unwise to choose the term susan for the constant to ground X with, it means that this would probably be a false inference. Obviously, it's possible that John would have a sister named Susan and a brother named Susan too, whether it is not likely. In fact, if we choose a totally new constant rather then, there can be no problems and the inference is guaranteed to be correct.