Consider the following premises. forall x_11: P_11(x_11) <-> W(x_11). forall x_11: P_11(x_11). forall x_11: P_5(x_11). forall x_11: P_5(x_11) -> P(x_11). Can we infer exists x_11: ~(P(x_11) and W(x_11)) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) <-> W(x_11) we can infer forall x_11: P_11(x_11) -> W(x_11) via universal biconditional elimination. From forall x_11: P_5(x_11) -> P(x_11), forall x_11: P_5(x_11) we can infer forall x_11: P(x_11) via universal modus ponens. From forall x_11: P_11(x_11) -> W(x_11), forall x_11: P_11(x_11) we can infer forall x_11: W(x_11) via universal modus ponens. Finally, from forall x_11: P(x_11), forall x_11: W(x_11) we can infer forall x_11: P(x_11) and W(x_11) via universal conjunction, which contradicts exists x_11: ~(P(x_11) and W(x_11)). Therefore, the answer is no.