Consider the following premises. forall x: W_2(x) -> Q(x). forall x: R_3(x) -> S_4(x). exists x: Q(x) -> S_4(x). forall x: ~R_3(x). forall x: S_4(x) <-> R_3(x). Can we infer forall x: W_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S_4(x) <-> R_3(x) we can infer forall x: S_4(x) -> R_3(x) via universal biconditional elimination. From forall x: W_2(x) -> Q(x), exists x: Q(x) -> S_4(x) we can infer exists x: W_2(x) -> S_4(x) via existential transitivity. From forall x: R_3(x) -> S_4(x), forall x: S_4(x) -> R_3(x) we can infer forall x: R_3(x) <-> S_4(x) via universal biconditional introduction. From forall x: R_3(x) <-> S_4(x), forall x: ~R_3(x) we can infer forall x: ~S_4(x) via universal biconditional elimination. Finally, from exists x: W_2(x) -> S_4(x), forall x: ~S_4(x) we can infer exists x: ~W_2(x) via existential modus tollens, which contradicts forall x: W_2(x). Therefore, the answer is no.