Consider the following premises. For all x_10, x_10 will drink water or x_10 is a nurse. For all x_10, x_10 will listen to a song if and only if x_10 will drink water. There is at least one x_10 for which if x_10 were a nurse, then x_10 is writing a letter. For all x_10, if x_10 writes a letter, then x_10 is a carpenter. For all x_10, if x_10 were a carpenter, then x_10 is listening to a song. Can we infer the following from them? There is at least one x_10 for which if x_10 were a nurse, then x_10 is writing a letter.
That is one of the premises. Therefore, the answer is yes.