Consider the following premises. forall x_11: P_11(x_11) or P_1(x_11). exists x_11: ~P_11(x_11). forall x_11: P_6(x_11) or Q_6(x_11). forall x_11: P_6(x_11) -> R_6(x_11). forall x_11: Q_6(x_11) -> R_6(x_11). Can we infer exists x_11: R_6(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) or P_1(x_11), exists x_11: ~P_11(x_11) we can infer exists x_11: P_1(x_11) via existential disjunctive syllogism. From forall x_11: P_6(x_11) -> R_6(x_11), forall x_11: Q_6(x_11) -> R_6(x_11), forall x_11: P_6(x_11) or Q_6(x_11) we can infer forall x_11: R_6(x_11) via universal disjunction elimination. From exists x_11: P_1(x_11), forall x_11: R_6(x_11) we can infer exists x_11: P_1(x_11) and R_6(x_11) via existential conjunction. Finally, from exists x_11: P_1(x_11) and R_6(x_11) we can infer exists x_11: R_6(x_11) via existential simplification. Therefore, the answer is yes.