Consider the following premises. forall x_8: P_3(x_8) -> W_2(x_8). forall x_8: W_2(x_8) -> ~Q_8(x_8). forall x_8: W_2(x_8) -> P_3(x_8). forall x_8: Q_8(x_8). forall x_8: W_2(x_8) -> P_3(x_8). Can we infer exists x_8: P_3(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: W_2(x_8) -> ~Q_8(x_8), forall x_8: Q_8(x_8) we can infer forall x_8: ~W_2(x_8) via universal modus tollens. From forall x_8: W_2(x_8) -> P_3(x_8), forall x_8: P_3(x_8) -> W_2(x_8) we can infer forall x_8: W_2(x_8) <-> P_3(x_8) via universal biconditional introduction. From forall x_8: W_2(x_8) <-> P_3(x_8) we can infer forall x_8: P_3(x_8) -> W_2(x_8) via universal biconditional elimination. From forall x_8: P_3(x_8) -> W_2(x_8), forall x_8: W_2(x_8) -> P_3(x_8) we can infer forall x_8: P_3(x_8) <-> W_2(x_8) via universal biconditional introduction. Finally, from forall x_8: P_3(x_8) <-> W_2(x_8), forall x_8: ~W_2(x_8) we can infer forall x_8: ~P_3(x_8) via universal biconditional elimination, which contradicts exists x_8: P_3(x_8). Therefore, the answer is no.