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