Consider the following premises. For all x_6, x_6 will write a letter or x_6 is happy. For all x_6, x_6 will make tea. For all x_6, x_6 will work. For all x_6, if x_6 writes a letter, then x_6 is not working. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which either x_6 is happy or x_6 will make tea are not true.
 From the fact that for all x_6, if x_6 writes a letter, then x_6 is not working, and that for all x_6, x_6 will work we can infer that for all x_6, x_6 will not write a letter via universal modus tollens. From the fact that for all x_6, x_6 will write a letter or x_6 is happy, and that for all x_6, x_6 will not write a letter we can infer that for all x_6, x_6 is happy via universal disjunctive syllogism. Finally, from the fact that for all x_6, x_6 is happy, and that for all x_6, x_6 will make tea we can infer that for all x_6, x_6 is happy and x_6 will make tea via universal conjunction, which contradicts that There is at least one x_6 for which either x_6 is happy or x_6 will make tea are not true. Therefore, the answer is no.