Consider the following premises. For all x_13, if x_13 were a mechanic, then x_13 is cooking. For all x_13, x_13 will cook if and only if x_13 will play squash. There is at least one x_13 for which x_13 is a mechanic or x_13 will play squash. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, neither x_13 will cook nor x_13 is a soldier.
 From the fact that for all x_13, x_13 will cook if and only if x_13 will play squash we can infer that for all x_13, if x_13 plays squash, then x_13 is cooking via universal biconditional elimination. From the fact that for all x_13, if x_13 were a mechanic, then x_13 is cooking, that for all x_13, if x_13 plays squash, then x_13 is cooking, and that there is at least one x_13 for which x_13 is a mechanic or x_13 will play squash we can infer that there is at least one x_13 for which x_13 will cook via existential disjunction elimination. Finally, from the fact that there is at least one x_13 for which x_13 will cook we can infer that there is at least one x_13 for which x_13 will cook or x_13 is a soldier via existential addition, which contradicts that For all x_13, neither x_13 will cook nor x_13 is a soldier. Therefore, the answer is no.