Consider the following premises. For all x_3, if x_3 were an astronaut, then x_3 is cooking. For all x_3, x_3 is an astronaut or x_3 will not cook. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which the fact that x_3 cooks does not imply that x_3 is an astronaut.
 From the fact that for all x_3, x_3 is an astronaut or x_3 will not cook we can infer that for all x_3, if x_3 cooks, then x_3 is an astronaut via universal rewriting. From the fact that for all x_3, if x_3 cooks, then x_3 is an astronaut, and that for all x_3, if x_3 were an astronaut, then x_3 is cooking we can infer that for all x_3, x_3 will cook if and only if x_3 is an astronaut via universal biconditional introduction. Finally, from the fact that for all x_3, x_3 will cook if and only if x_3 is an astronaut we can infer that for all x_3, if x_3 cooks, then x_3 is an astronaut via universal biconditional elimination, which contradicts that There is at least one x_3 for which the fact that x_3 cooks does not imply that x_3 is an astronaut. Therefore, the answer is no.