Consider the following premises. For all x, if x writes a letter, then x is playing squash. For all x, if x plays squash, then x is writing a letter. For all x, if x were an artist, then x is writing a letter. There is at least one x for which if x writes a letter, then x is an artist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which if x were an artist, then x is playing squash.
 From the fact that for all x, if x writes a letter, then x is playing squash, and that for all x, if x plays squash, then x is writing a letter we can infer that for all x, x will write a letter if and only if x will play squash via universal biconditional introduction. From the fact that for all x, x will write a letter if and only if x will play squash we can infer that for all x, if x writes a letter, then x is playing squash via universal biconditional elimination. From the fact that for all x, if x were an artist, then x is writing a letter, and that there is at least one x for which if x writes a letter, then x is an artist we can infer that there is at least one x for which x is an artist if and only if x will write a letter via existential biconditional introduction. From the fact that there is at least one x for which x is an artist if and only if x will write a letter we can infer that there is at least one x for which if x were an artist, then x is writing a letter via existential biconditional elimination. Finally, from the fact that there is at least one x for which if x were an artist, then x is writing a letter, and that for all x, if x writes a letter, then x is playing squash we can infer that there is at least one x for which if x were an artist, then x is playing squash via existential transitivity. Therefore, the answer is yes.