Consider the following premises. forall x: P_2(x) <-> U(x). forall x: ~R(x). forall x: U(x) -> R(x). forall x: R(x) -> U(x). Can we infer forall x: ~P_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) -> R(x), forall x: R(x) -> U(x) we can infer forall x: U(x) <-> R(x) via universal biconditional introduction. From forall x: U(x) <-> R(x), forall x: ~R(x) we can infer forall x: ~U(x) via universal biconditional elimination. Finally, from forall x: P_2(x) <-> U(x), forall x: ~U(x) we can infer forall x: ~P_2(x) via universal biconditional elimination. Therefore, the answer is yes.