Consider the following premises. forall x_2: S_2(x_2) <-> U(x_2). forall x_2: U(x_2) or W(x_2). forall x_2: W(x_2) -> S_2(x_2). forall x_2: P_3(x_2). Can we infer exists x_2: ~(S_2(x_2) and P_3(x_2)) from them? If possible, name the inference rules being used at each step.
 From forall x_2: S_2(x_2) <-> U(x_2) we can infer forall x_2: U(x_2) -> S_2(x_2) via universal biconditional elimination. From forall x_2: U(x_2) -> S_2(x_2), forall x_2: W(x_2) -> S_2(x_2), forall x_2: U(x_2) or W(x_2) we can infer forall x_2: S_2(x_2) via universal disjunction elimination. Finally, from forall x_2: S_2(x_2), forall x_2: P_3(x_2) we can infer forall x_2: S_2(x_2) and P_3(x_2) via universal conjunction, which contradicts exists x_2: ~(S_2(x_2) and P_3(x_2)). Therefore, the answer is no.