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