Consider the following premises. For all x_2, if x_2 cooks, then x_2 is playing tennis. For all x_2, if x_2 were a lawyer, then x_2 is cooking. For all x_2, x_2 is a lawyer if and only if x_2 will climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x_2, if x_2 climbs a mountain, then x_2 is playing tennis.
 From the fact that for all x_2, if x_2 were a lawyer, then x_2 is cooking, and that for all x_2, if x_2 cooks, then x_2 is playing tennis we can infer that for all x_2, if x_2 were a lawyer, then x_2 is playing tennis via universal transitivity. From the fact that for all x_2, x_2 is a lawyer if and only if x_2 will climb a mountain we can infer that for all x_2, if x_2 climbs a mountain, then x_2 is a lawyer via universal biconditional elimination. Finally, from the fact that for all x_2, if x_2 climbs a mountain, then x_2 is a lawyer, and that for all x_2, if x_2 were a lawyer, then x_2 is playing tennis we can infer that for all x_2, if x_2 climbs a mountain, then x_2 is playing tennis via universal transitivity. Therefore, the answer is yes.