Consider the following premises. For all x_11, x_11 is fast if and only if x_11 will play a game. For all x_11, x_11 is a mechanic if and only if x_11 will play a game. For all x_11, x_11 is fast or x_11 is a mechanic. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_11 for which neither x_11 will play a game nor x_11 will listen to a song.
 From the fact that for all x_11, x_11 is fast if and only if x_11 will play a game we can infer that for all x_11, if x_11 were fast, then x_11 is playing a game via universal biconditional elimination. From the fact that for all x_11, x_11 is a mechanic if and only if x_11 will play a game we can infer that for all x_11, if x_11 were a mechanic, then x_11 is playing a game via universal biconditional elimination. From the fact that for all x_11, if x_11 were fast, then x_11 is playing a game, that for all x_11, if x_11 were a mechanic, then x_11 is playing a game, and that for all x_11, x_11 is fast or x_11 is a mechanic we can infer that for all x_11, x_11 will play a game via universal disjunction elimination. Finally, from the fact that for all x_11, x_11 will play a game we can infer that for all x_11, x_11 will play a game or x_11 will listen to a song via universal addition, which contradicts that There is at least one x_11 for which neither x_11 will play a game nor x_11 will listen to a song. Therefore, the answer is no.