Consider the following premises. forall x_11: P_11(x_11) -> Q_11(x_11). forall x_11: P_10(x_11) or P_11(x_11). forall x_11: R_10(x_11) -> W(x_11). forall x_11: P_6(x_11) or ~W(x_11). forall x_11: ~P_10(x_11) or R_10(x_11). forall x_11: Q_11(x_11) -> W(x_11). Can we infer exists x_11: ~P_6(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) -> Q_11(x_11), forall x_11: Q_11(x_11) -> W(x_11) we can infer forall x_11: P_11(x_11) -> W(x_11) via universal transitivity. From forall x_11: P_10(x_11) or P_11(x_11), forall x_11: ~P_10(x_11) or R_10(x_11) we can infer forall x_11: P_11(x_11) or R_10(x_11) via universal resolution. From forall x_11: P_6(x_11) or ~W(x_11) we can infer forall x_11: W(x_11) -> P_6(x_11) via universal rewriting. From forall x_11: P_11(x_11) -> W(x_11), forall x_11: R_10(x_11) -> W(x_11), forall x_11: P_11(x_11) or R_10(x_11) we can infer forall x_11: W(x_11) via universal disjunction elimination. Finally, from forall x_11: W(x_11) -> P_6(x_11), forall x_11: W(x_11) we can infer forall x_11: P_6(x_11) via universal modus ponens, which contradicts exists x_11: ~P_6(x_11). Therefore, the answer is no.