Consider the following premises. For all y_1, y_1 will write a letter if and only if y_1 is happy. There is at least one y_1 for which y_1 will write a letter if and only if y_1 is a nurse. Can we infer the following from them? There is at least one y_1 for which y_1 will write a letter if and only if y_1 is a nurse.
That is one of the premises. Therefore, the answer is yes.