Consider the following premises. For all x_11, x_11 will listen to a song if and only if x_11 will play tennis. There is at least one x_11 for which x_11 will play tennis. For all x_11, x_11 will not listen to a song. 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 x_11 is happy.
 From the fact that for all x_11, x_11 will listen to a song if and only if x_11 will play tennis we can infer that for all x_11, if x_11 plays tennis, then x_11 is listening to a song via universal biconditional elimination. From the fact that for all x_11, if x_11 plays tennis, then x_11 is listening to a song, and that for all x_11, x_11 will not listen to a song we can infer that for all x_11, x_11 will not play tennis via universal modus tollens. From the fact that there is at least one x_11 for which x_11 will play tennis we can infer that there is at least one x_11 for which x_11 will play tennis or x_11 is happy via existential addition. Finally, from the fact that there is at least one x_11 for which x_11 will play tennis or x_11 is happy, and that for all x_11, x_11 will not play tennis we can infer that there is at least one x_11 for which x_11 is happy via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.