Consider the following premises. forall x: R(x) -> P_2(x). forall x: P_2(x) -> R(x). forall x: R(x) <-> P_2(x). Can we infer exists x: ~(P_2(x) <-> R(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: P_2(x) -> R(x), forall x: R(x) -> P_2(x) we can infer forall x: P_2(x) <-> R(x) via universal biconditional introduction. From forall x: R(x) <-> P_2(x) we can infer forall x: P_2(x) -> R(x) via universal biconditional elimination. From forall x: P_2(x) <-> R(x) we can infer forall x: R(x) -> P_2(x) via universal biconditional elimination. Finally, from forall x: P_2(x) -> R(x), forall x: R(x) -> P_2(x) we can infer forall x: P_2(x) <-> R(x) via universal biconditional introduction, which contradicts exists x: ~(P_2(x) <-> R(x)). Therefore, the answer is no.