Consider the following premises. For all x_9, x_9 is a lawyer if and only if x_9 will cook. For all x_9, x_9 is not a carpenter. For all x_9, x_9 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: For all x_9, x_9 is a lawyer.
 From the fact that for all x_9, x_9 is a carpenter we can infer that for all x_9, x_9 is a carpenter or x_9 will cook via universal addition. From the fact that for all x_9, x_9 is a carpenter or x_9 will cook, and that for all x_9, x_9 is not a carpenter we can infer that for all x_9, x_9 will cook via universal disjunctive syllogism. From the fact that for all x_9, x_9 is a lawyer if and only if x_9 will cook we can infer that for all x_9, if x_9 cooks, then x_9 is a lawyer via universal biconditional elimination. Finally, from the fact that for all x_9, if x_9 cooks, then x_9 is a lawyer, and that for all x_9, x_9 will cook we can infer that for all x_9, x_9 is a lawyer via universal modus ponens. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.