Consider the following premises. There is at least one x_13 for which x_13 will not sleep. For all x_13, x_13 will sleep or x_13 will climb a mountain. For all x_13, if x_13 were a musician, then x_13 is playing a game. For all x_13, if x_13 were sad, then x_13 is playing a game. For all x_13, x_13 is a musician or x_13 is sad. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, either x_13 will play a game or x_13 will climb a mountain are not true.
 From the fact that for all x_13, if x_13 were a musician, then x_13 is playing a game, that for all x_13, if x_13 were sad, then x_13 is playing a game, and that for all x_13, x_13 is a musician or x_13 is sad we can infer that for all x_13, x_13 will play a game via universal disjunction elimination. From the fact that for all x_13, x_13 will sleep or x_13 will climb a mountain, and that there is at least one x_13 for which x_13 will not sleep we can infer that there is at least one x_13 for which x_13 will climb a mountain via existential disjunctive syllogism. Finally, from the fact that for all x_13, x_13 will play a game, and that there is at least one x_13 for which x_13 will climb a mountain we can infer that there is at least one x_13 for which x_13 will play a game and x_13 will climb a mountain via existential conjunction, which contradicts that For all x_13, either x_13 will play a game or x_13 will climb a mountain are not true. Therefore, the answer is no.