Consider the following premises. For all x_7, x_7 is a poet if and only if x_7 is a carpenter. For all x_7, x_7 is a carpenter or x_7 will cook. There is at least one x_7 for which x_7 is a poet or x_7 will not cook. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 is not a poet.
 From the fact that there is at least one x_7 for which x_7 is a poet or x_7 will not cook we can infer that there is at least one x_7 for which if x_7 cooks, then x_7 is a poet via existential rewriting. From the fact that for all x_7, x_7 is a poet if and only if x_7 is a carpenter we can infer that for all x_7, if x_7 were a carpenter, then x_7 is a poet via universal biconditional elimination. Finally, from the fact that for all x_7, if x_7 were a carpenter, then x_7 is a poet, that there is at least one x_7 for which if x_7 cooks, then x_7 is a poet, and that for all x_7, x_7 is a carpenter or x_7 will cook we can infer that there is at least one x_7 for which x_7 is a poet via existential disjunction elimination, which contradicts that For all x_7, x_7 is not a poet. Therefore, the answer is no.