Consider the following premises. For all x_10, x_10 is excited if and only if x_10 will go running. There is at least one x_10 for which x_10 is excited and x_10 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_10 for which x_10 will go running.
 From the fact that there is at least one x_10 for which x_10 is excited and x_10 will climb a mountain we can infer that there is at least one x_10 for which x_10 is excited via existential simplification. From the fact that for all x_10, x_10 is excited if and only if x_10 will go running we can infer that for all x_10, if x_10 were excited, then x_10 is running via universal biconditional elimination. Finally, from the fact that for all x_10, if x_10 were excited, then x_10 is running, and that there is at least one x_10 for which x_10 is excited we can infer that there is at least one x_10 for which x_10 will go running via existential modus ponens. Therefore, the answer is yes.