Consider the following premises. For all x_6, if x_6 climbs a mountain, then x_6 is cooking. There is at least one x_6 for which x_6 is happy or x_6 will climb a mountain. For all x_6, x_6 will cook or x_6 will not make tea. For all x_6, if x_6 were happy, then x_6 is making tea. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will cook.
 From the fact that for all x_6, x_6 will cook or x_6 will not make tea we can infer that for all x_6, if x_6 makes tea, then x_6 is cooking via universal rewriting. From the fact that for all x_6, if x_6 were happy, then x_6 is making tea, and that for all x_6, if x_6 makes tea, then x_6 is cooking we can infer that for all x_6, if x_6 were happy, then x_6 is cooking via universal transitivity. Finally, from the fact that for all x_6, if x_6 were happy, then x_6 is cooking, that for all x_6, if x_6 climbs a mountain, then x_6 is cooking, and that there is at least one x_6 for which x_6 is happy or x_6 will climb a mountain we can infer that there is at least one x_6 for which x_6 will cook via existential disjunction elimination. Therefore, the answer is yes.