Consider the following premises. There is at least one x_4 for which x_4 will go running if and only if x_4 will play a game. For all x_4, x_4 will not go running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which x_4 will not play a game.
We can infer this via the existential biconditional elimination rule. Therefore, the answer is yes.