Consider the following premises. forall x_7: P(x_7) -> W_1(x_7). forall x_7: ~P_7(x_7) or P_2(x_7). forall x_7: P_7(x_7) or P(x_7). exists x_7: P_2(x_7) <-> W_1(x_7). Can we infer forall x_7: ~W_1(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) or P(x_7), forall x_7: ~P_7(x_7) or P_2(x_7) we can infer forall x_7: P(x_7) or P_2(x_7) via universal resolution. From exists x_7: P_2(x_7) <-> W_1(x_7) we can infer exists x_7: P_2(x_7) -> W_1(x_7) via existential biconditional elimination. Finally, from forall x_7: P(x_7) -> W_1(x_7), exists x_7: P_2(x_7) -> W_1(x_7), forall x_7: P(x_7) or P_2(x_7) we can infer exists x_7: W_1(x_7) via existential disjunction elimination, which contradicts forall x_7: ~W_1(x_7). Therefore, the answer is no.