Consider the following premises. For all x_15, x_15 is sad if and only if x_15 will play squash. There is at least one x_15 for which x_15 will play squash. For all x_15, x_15 will play a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_15, x_15 is not sad.
 From the fact that for all x_15, x_15 is sad if and only if x_15 will play squash we can infer that for all x_15, if x_15 plays squash, then x_15 is sad via universal biconditional elimination. From the fact that for all x_15, if x_15 plays squash, then x_15 is sad, and that there is at least one x_15 for which x_15 will play squash we can infer that there is at least one x_15 for which x_15 is sad via existential modus ponens. From the fact that for all x_15, x_15 will play a game, and that there is at least one x_15 for which x_15 is sad we can infer that there is at least one x_15 for which x_15 will play a game and x_15 is sad via existential conjunction. Finally, from the fact that there is at least one x_15 for which x_15 will play a game and x_15 is sad we can infer that there is at least one x_15 for which x_15 will play a game via existential simplification, which contradicts that For all x_15, x_15 is not sad. Therefore, the answer is no.