Consider the following premises. exists x_12: Q_4(x_12). forall x_12: Q_4(x_12) <-> U_2(x_12). Can we infer forall x_12: ~(U_2(x_12) or W_1(x_12)) from them? If possible, name the inference rules being used at each step.
 From forall x_12: Q_4(x_12) <-> U_2(x_12) we can infer forall x_12: Q_4(x_12) -> U_2(x_12) via universal biconditional elimination. From forall x_12: Q_4(x_12) -> U_2(x_12), exists x_12: Q_4(x_12) we can infer exists x_12: U_2(x_12) via existential modus ponens. Finally, from exists x_12: U_2(x_12) we can infer exists x_12: U_2(x_12) or W_1(x_12) via existential addition, which contradicts forall x_12: ~(U_2(x_12) or W_1(x_12)). Therefore, the answer is no.