Consider the following premises. For all x, if x climbs a mountain, then x is fast. There is at least one x for which if x goes running, then x is an artist. For all x, x will go running or x is not fast. For all x, x will climb a mountain if and only if 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 x is an artist if and only if x is fast.
 From the fact that for all x, x will go running or x is not fast we can infer that for all x, if x were fast, then x is running via universal rewriting. From the fact that for all x, x will climb a mountain if and only if x is an artist we can infer that for all x, if x were an artist, then x is climbing a mountain via universal biconditional elimination. From the fact that for all x, if x were an artist, then x is climbing a mountain, and that for all x, if x climbs a mountain, then x is fast we can infer that for all x, if x were an artist, then x is fast via universal transitivity. From the fact that for all x, if x were fast, then x is running, and that there is at least one x for which if x goes running, then x is an artist we can infer that there is at least one x for which if x were fast, then x is an artist via existential transitivity. Finally, from the fact that for all x, if x were an artist, then x is fast, and that there is at least one x for which if x were fast, 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 is fast via existential biconditional introduction. Therefore, the answer is yes.