Consider the following premises. For all x_5, x_5 will work or x_5 will not make tea. For all x_5, if x_5 were a writer, then x_5 is cooking. There is at least one x_5 for which if x_5 works, then x_5 is a writer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which if x_5 makes tea, then x_5 is cooking.
 From the fact that there is at least one x_5 for which if x_5 works, then x_5 is a writer, and that for all x_5, if x_5 were a writer, then x_5 is cooking we can infer that there is at least one x_5 for which if x_5 works, then x_5 is cooking via existential transitivity. From the fact that for all x_5, x_5 will work or x_5 will not make tea we can infer that for all x_5, if x_5 makes tea, then x_5 is working via universal rewriting. Finally, from the fact that for all x_5, if x_5 makes tea, then x_5 is working, and that there is at least one x_5 for which if x_5 works, then x_5 is cooking we can infer that there is at least one x_5 for which if x_5 makes tea, then x_5 is cooking via existential transitivity. Therefore, the answer is yes.