Consider the following premises. forall x: W_4(x) -> R(x). forall x: R(x) -> Q(x). forall x: R_2(x) -> W_4(x). forall x: Q(x) -> R_2(x). Can we infer exists x: ~(R_2(x) -> R(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: R_2(x) -> W_4(x), forall x: W_4(x) -> R(x) we can infer forall x: R_2(x) -> R(x) via universal transitivity. From forall x: R(x) -> Q(x), forall x: Q(x) -> R_2(x) we can infer forall x: R(x) -> R_2(x) via universal transitivity. From forall x: R(x) -> R_2(x), forall x: R_2(x) -> R(x) we can infer forall x: R(x) <-> R_2(x) via universal biconditional introduction. Finally, from forall x: R(x) <-> R_2(x) we can infer forall x: R_2(x) -> R(x) via universal biconditional elimination, which contradicts exists x: ~(R_2(x) -> R(x)). Therefore, the answer is no.