Consider the following premises. forall x_10: T_3(x_10). forall x_10: T_3(x_10) -> Q(x_10). forall x_10: P_10(x_10) or R(x_10). forall x_10: R(x_10) -> W(x_10). forall x_10: U_3(x_10) -> W(x_10). forall x_10: ~P_10(x_10) or U_3(x_10). Can we infer exists x_10: ~(W(x_10) and Q(x_10)) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) or R(x_10), forall x_10: ~P_10(x_10) or U_3(x_10) we can infer forall x_10: R(x_10) or U_3(x_10) via universal resolution. From forall x_10: R(x_10) -> W(x_10), forall x_10: U_3(x_10) -> W(x_10), forall x_10: R(x_10) or U_3(x_10) we can infer forall x_10: W(x_10) via universal disjunction elimination. From forall x_10: T_3(x_10) -> Q(x_10), forall x_10: T_3(x_10) we can infer forall x_10: Q(x_10) via universal modus ponens. Finally, from forall x_10: W(x_10), forall x_10: Q(x_10) we can infer forall x_10: W(x_10) and Q(x_10) via universal conjunction, which contradicts exists x_10: ~(W(x_10) and Q(x_10)). Therefore, the answer is no.