Consider the following premises. There is at least one x_7 for which x_7 will climb a mountain if and only if x_7 will go running. For all x_7, x_7 will not climb a mountain. For all x_7, x_7 will go running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 will write a letter.
 From the fact that there is at least one x_7 for which x_7 will climb a mountain if and only if x_7 will go running, and that for all x_7, x_7 will not climb a mountain we can infer that there is at least one x_7 for which x_7 will not go running via existential biconditional elimination. From the fact that for all x_7, x_7 will go running we can infer that for all x_7, x_7 will go running or x_7 will write a letter via universal addition. Finally, from the fact that for all x_7, x_7 will go running or x_7 will write a letter, and that there is at least one x_7 for which x_7 will not go running we can infer that there is at least one x_7 for which x_7 will write a letter via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.