Consider the following premises. Patricia is a doctor if and only if James is listening to a song. Patricia is not rich. For all x_8, if x_8 were a doctor, then x_8 is rich. For all x_8, if x_8 were rich, then x_8 is a doctor. Can we infer the following from them? For all x_8, if x_8 were a doctor, then x_8 is rich.
That is one of the premises. Therefore, the answer is yes.