Consider the following premises. If Karen plays a game, then Thomas is not a mechanic. For all x, x will play a game if and only if x will listen to a song. Thomas is a mechanic. Patricia is playing squash if and only if Karen is listening to a song. Can we infer the following from them? If we can, name the inference rule being used: Patricia is playing squash.
 From the fact that for all x, x will play a game if and only if x will listen to a song we can infer that Karen is playing a game if and only if Karen is listening to a song via universal instantiation. From the fact that if Karen plays a game, then Thomas is not a mechanic, and that Thomas is a mechanic we can infer that Karen is not playing a game via modus tollens. From the fact that Karen is playing a game if and only if Karen is listening to a song we can infer that if Karen listens to a song, then Karen is playing a game via biconditional elimination. From the fact that if Karen listens to a song, then Karen is playing a game, and that Karen is not playing a game we can infer that Karen is not listening to a song via modus tollens. Finally, from the fact that Patricia is playing squash if and only if Karen is listening to a song, and that Karen is not listening to a song we can infer that Patricia is not playing squash via biconditional elimination, which contradicts that Patricia is playing squash. Therefore, the answer is no.