Consider the following premises. forall x_12: P_12(x_12) <-> W_1(x_12). forall x_12: P_12(x_12) <-> S(x_12). exists x_12: W_1(x_12). Can we infer forall x_12: ~S(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) <-> W_1(x_12) we can infer forall x_12: W_1(x_12) -> P_12(x_12) via universal biconditional elimination. From forall x_12: P_12(x_12) <-> S(x_12) we can infer forall x_12: P_12(x_12) -> S(x_12) via universal biconditional elimination. From forall x_12: W_1(x_12) -> P_12(x_12), forall x_12: P_12(x_12) -> S(x_12) we can infer forall x_12: W_1(x_12) -> S(x_12) via universal transitivity. Finally, from forall x_12: W_1(x_12) -> S(x_12), exists x_12: W_1(x_12) we can infer exists x_12: S(x_12) via existential modus ponens, which contradicts forall x_12: ~S(x_12). Therefore, the answer is no.