Consider the following premises. There is at least one x for which x is a politician if and only if x is a nurse. For all x, if x were a cashier, then x is a nurse. For all x, if x were a nurse, then x is a cashier. Can we infer the following from them? There is at least one x for which x is a politician if and only if x is a nurse.
That is one of the premises. Therefore, the answer is yes.