Consider the following premises. There is at least one x_10 for which x_10 will work or x_10 will not climb a mountain. For all x_10, if x_10 works, then x_10 is writing a letter. For all x_10, if x_10 works, then x_10 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which if x_10 climbs a mountain, then x_10 is writing a letter.
 From the fact that there is at least one x_10 for which x_10 will work or x_10 will not climb a mountain we can infer that there is at least one x_10 for which if x_10 climbs a mountain, then x_10 is working via existential rewriting. From the fact that there is at least one x_10 for which if x_10 climbs a mountain, then x_10 is working, and that for all x_10, if x_10 works, then x_10 is climbing a mountain we can infer that there is at least one x_10 for which x_10 will climb a mountain if and only if x_10 will work via existential biconditional introduction. From the fact that there is at least one x_10 for which x_10 will climb a mountain if and only if x_10 will work we can infer that there is at least one x_10 for which if x_10 climbs a mountain, then x_10 is working via existential biconditional elimination. Finally, from the fact that there is at least one x_10 for which if x_10 climbs a mountain, then x_10 is working, and that for all x_10, if x_10 works, then x_10 is writing a letter we can infer that there is at least one x_10 for which if x_10 climbs a mountain, then x_10 is writing a letter via existential transitivity. Therefore, the answer is yes.