Consider the following premises. For all x_6, if x_6 were a mechanic, then x_6 is not bored. For all x_6, if x_6 writes a letter, then x_6 is a mechanic. There is at least one x_6 for which if x_6 cooks, then x_6 is writing a letter. For all x_6, x_6 is bored. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will cook.
 From the fact that for all x_6, if x_6 were a mechanic, then x_6 is not bored, and that for all x_6, x_6 is bored we can infer that for all x_6, x_6 is not a mechanic via universal modus tollens. From the fact that for all x_6, if x_6 writes a letter, then x_6 is a mechanic, and that for all x_6, x_6 is not a mechanic we can infer that for all x_6, x_6 will not write a letter via universal modus tollens. Finally, from the fact that there is at least one x_6 for which if x_6 cooks, then x_6 is writing a letter, and that for all x_6, x_6 will not write a letter we can infer that there is at least one x_6 for which x_6 will not cook via existential modus tollens, which contradicts that For all x_6, x_6 will cook. Therefore, the answer is no.