Consider the following premises. There is at least one x for which if x were a writer, then x is climbing a mountain. For all x, if x climbs a mountain, then x is listening to a song. For all x, if x listens to a song, then x is a writer. Can we infer the following from them? There is at least one x for which if x were a writer, then x is climbing a mountain.
That is one of the premises. Therefore, the answer is yes.