Consider the following premises. There is at least one y for which if y climbs a mountain, then y is running. For all y, if y plays tennis, then y is climbing a mountain. For all y, if y goes running, then y is playing tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which if y climbs a mountain, then y is playing tennis.
We can infer this via the existential transitivity rule. Therefore, the answer is yes.