Consider the following premises. There is at least one x for which x is fast or x will not go running. For all x, if x were fast, then x is running. For all x, if x goes running, then x is listening to a song. For all x, x will not listen to a song. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is not fast.
 From the fact that for all x, if x goes running, then x is listening to a song, and that for all x, x will not listen to a song we can infer that for all x, x will not go running via universal modus tollens. From the fact that there is at least one x for which x is fast or x will not go running we can infer that there is at least one x for which if x goes running, then x is fast via existential rewriting. From the fact that for all x, if x were fast, then x is running, and that there is at least one x for which if x goes running, then x is fast we can infer that there is at least one x for which x is fast if and only if x will go running via existential biconditional introduction. Finally, from the fact that there is at least one x for which x is fast if and only if x will go running, and that for all x, x will not go running we can infer that there is at least one x for which x is not fast via existential biconditional elimination. Therefore, the answer is yes.