Consider the following premises. forall x_11: P_11(x_11) <-> P_1(x_11). exists x_11: P(x_11) or W_2(x_11). forall x_11: ~P_11(x_11). forall x_11: P_1(x_11) or ~P(x_11). Can we infer forall x_11: ~W_2(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) <-> P_1(x_11), forall x_11: ~P_11(x_11) we can infer forall x_11: ~P_1(x_11) via universal biconditional elimination. From forall x_11: P_1(x_11) or ~P(x_11) we can infer forall x_11: P(x_11) -> P_1(x_11) via universal rewriting. From forall x_11: P(x_11) -> P_1(x_11), forall x_11: ~P_1(x_11) we can infer forall x_11: ~P(x_11) via universal modus tollens. Finally, from exists x_11: P(x_11) or W_2(x_11), forall x_11: ~P(x_11) we can infer exists x_11: W_2(x_11) via existential disjunctive syllogism, which contradicts forall x_11: ~W_2(x_11). Therefore, the answer is no.