Consider the following premises. For all x, if x plays squash, then x is playing a game. There is at least one x for which x will play squash if and only if x will write a letter. For all x, if x plays squash, then x is writing a letter. 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 writes a letter, then x is playing a game.
 From the fact that there is at least one x for which x will play squash if and only if x will write a letter we can infer that there is at least one x for which if x writes a letter, then x is playing squash via existential biconditional elimination. From the fact that for all x, if x plays squash, then x is writing a letter, and that there is at least one x for which if x writes a letter, then x is playing squash we can infer that there is at least one x for which x will play squash 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 will play squash if and only if x will write a letter we can infer that there is at least one x for which if x writes a letter, then x is playing squash via existential biconditional elimination. Finally, from the fact that there is at least one x for which if x writes a letter, then x is playing squash, and that for all x, if x plays squash, then x is playing a game we can infer that there is at least one x for which if x writes a letter, then x is playing a game via existential transitivity. Therefore, the answer is yes.