Consider the following premises. For all x_13, if x_13 were bored, then x_13 is running. For all x_13, x_13 is excited if and only if x_13 will go running. For all x_13, x_13 will not listen to a song. There is at least one x_13 for which if x_13 climbs a mountain, then x_13 is bored. For all x_13, x_13 will listen to a song or x_13 will climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_13 for which x_13 will go running.
 From the fact that for all x_13, x_13 will listen to a song or x_13 will climb a mountain, and that for all x_13, x_13 will not listen to a song we can infer that for all x_13, x_13 will climb a mountain via universal disjunctive syllogism. From the fact that there is at least one x_13 for which if x_13 climbs a mountain, then x_13 is bored, and that for all x_13, if x_13 were bored, then x_13 is running we can infer that there is at least one x_13 for which if x_13 climbs a mountain, then x_13 is running via existential transitivity. From the fact that for all x_13, x_13 will climb a mountain we can infer that for all x_13, x_13 will climb a mountain or x_13 is excited via universal addition. From the fact that for all x_13, x_13 is excited if and only if x_13 will go running we can infer that for all x_13, if x_13 were excited, then x_13 is running via universal biconditional elimination. Finally, from the fact that there is at least one x_13 for which if x_13 climbs a mountain, then x_13 is running, that for all x_13, if x_13 were excited, then x_13 is running, and that for all x_13, x_13 will climb a mountain or x_13 is excited we can infer that there is at least one x_13 for which x_13 will go running via existential disjunction elimination. Therefore, the answer is yes.