Consider the following premises. For all x_10, x_10 is a doctor or x_10 will not play a game. For all x_10, if x_10 plays a game, then x_10 is a doctor. There is at least one x_10 for which if x_10 were a doctor, then x_10 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is a doctor if and only if x_10 will play a game.
We can infer this via the existential biconditional introduction rule. Therefore, the answer is yes.