Consider the following premises. For all x_12, x_12 will make tea if and only if x_12 is an astronaut. For all x_12, if x_12 makes tea, then x_12 is cooking. For all x_12, x_12 will play squash if and only if x_12 will cook. For all x_12, x_12 will play squash or x_12 is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_12 for which x_12 will not cook.
 From the fact that for all x_12, x_12 will make tea if and only if x_12 is an astronaut we can infer that for all x_12, if x_12 were an astronaut, then x_12 is making tea via universal biconditional elimination. From the fact that for all x_12, x_12 will play squash if and only if x_12 will cook we can infer that for all x_12, if x_12 plays squash, then x_12 is cooking via universal biconditional elimination. From the fact that for all x_12, if x_12 were an astronaut, then x_12 is making tea, and that for all x_12, if x_12 makes tea, then x_12 is cooking we can infer that for all x_12, if x_12 were an astronaut, then x_12 is cooking via universal transitivity. Finally, from the fact that for all x_12, if x_12 plays squash, then x_12 is cooking, that for all x_12, if x_12 were an astronaut, then x_12 is cooking, and that for all x_12, x_12 will play squash or x_12 is an astronaut we can infer that for all x_12, x_12 will cook via universal disjunction elimination, which contradicts that There is at least one x_12 for which x_12 will not cook. Therefore, the answer is no.