Consider the following premises. For all x, if x goes running, then x is cooking. For all x, if x plays squash, then x is cooking. For all x, if x cooks, then x is playing squash. For all x, x will go running or x will play squash. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which neither x will cook nor x will listen to a song.
 From the fact that for all x, if x cooks, then x is playing squash, and that for all x, if x plays squash, then x is cooking we can infer that for all x, x will cook if and only if x will play squash via universal biconditional introduction. From the fact that for all x, x will cook if and only if x will play squash we can infer that for all x, if x plays squash, then x is cooking via universal biconditional elimination. From the fact that for all x, if x goes running, then x is cooking, that for all x, if x plays squash, then x is cooking, and that for all x, x will go running or x will play squash we can infer that for all x, x will cook via universal disjunction elimination. Finally, from the fact that for all x, x will cook we can infer that for all x, x will cook or x will listen to a song via universal addition, which contradicts that There is at least one x for which neither x will cook nor x will listen to a song. Therefore, the answer is no.