Here is the question I have: “There exists a restaurant that, if it is midnight, then this restaurant would not be open.” and “Not all restaurant are open if it is midnight.” Assume the set of all restaurant as the universe of discourse.

I assume P(x) is the case that the restaurant will not be open and A is the event that it is midnight.

And I got $\exists x(A\to P(x))$ for both sentences. Is there anything I did wrong? Since the question assumes there will be two individual answers for the statements, and have a sub-question requiring me to prove these two logic notation to be logically equivalent.