Consider the following premises. There is at least one x_14 for which if x_14 were a poet, then x_14 is climbing a mountain. For all x_14, if x_14 plays a game, then x_14 is cooking. For all x_14, if x_14 cooks, then x_14 is a poet. For all x_14, x_14 will go running or x_14 will not climb a mountain. For all x_14, if x_14 goes running, then x_14 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_14, if is not true that x_14 will cook if and only if x_14 is a poet.
 From the fact that for all x_14, if x_14 goes running, then x_14 is playing a game, and that for all x_14, if x_14 plays a game, then x_14 is cooking we can infer that for all x_14, if x_14 goes running, then x_14 is cooking via universal transitivity. From the fact that for all x_14, x_14 will go running or x_14 will not climb a mountain we can infer that for all x_14, if x_14 climbs a mountain, then x_14 is running via universal rewriting. From the fact that for all x_14, if x_14 climbs a mountain, then x_14 is running, and that for all x_14, if x_14 goes running, then x_14 is cooking we can infer that for all x_14, if x_14 climbs a mountain, then x_14 is cooking via universal transitivity. From the fact that there is at least one x_14 for which if x_14 were a poet, then x_14 is climbing a mountain, and that for all x_14, if x_14 climbs a mountain, then x_14 is cooking we can infer that there is at least one x_14 for which if x_14 were a poet, then x_14 is cooking via existential transitivity. Finally, from the fact that for all x_14, if x_14 cooks, then x_14 is a poet, and that there is at least one x_14 for which if x_14 were a poet, then x_14 is cooking we can infer that there is at least one x_14 for which x_14 will cook if and only if x_14 is a poet via existential biconditional introduction, which contradicts that For all x_14, if is not true that x_14 will cook if and only if x_14 is a poet. Therefore, the answer is no.