Consider the following premises. forall x_2: Q_2(x_2) -> P(x_2). exists x_2: ~P(x_2). forall x_2: W_2(x_2) -> Q_2(x_2). forall x_2: P(x_2) -> W_2(x_2). Can we infer exists x_2: ~W_2(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: W_2(x_2) -> Q_2(x_2), forall x_2: Q_2(x_2) -> P(x_2) we can infer forall x_2: W_2(x_2) -> P(x_2) via universal transitivity. From forall x_2: W_2(x_2) -> P(x_2), forall x_2: P(x_2) -> W_2(x_2) we can infer forall x_2: W_2(x_2) <-> P(x_2) via universal biconditional introduction. Finally, from forall x_2: W_2(x_2) <-> P(x_2), exists x_2: ~P(x_2) we can infer exists x_2: ~W_2(x_2) via existential biconditional elimination. Therefore, the answer is yes.