Consider the following premises. forall x_7: P_7(x_7) or Q_7(x_7). forall x_7: Q_7(x_7) -> W_4(x_7). forall x_7: ~P_7(x_7) or R_7(x_7). forall x_7: R_7(x_7) -> W_4(x_7). Can we infer exists y_2: W_4(y_2) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) or Q_7(x_7), forall x_7: ~P_7(x_7) or R_7(x_7) we can infer forall x_7: Q_7(x_7) or R_7(x_7) via universal resolution. From forall x_7: Q_7(x_7) -> W_4(x_7), forall x_7: R_7(x_7) -> W_4(x_7), forall x_7: Q_7(x_7) or R_7(x_7) we can infer forall x_7: W_4(x_7) via universal disjunction elimination. From forall x_7: W_4(x_7) we can infer W_4(d_1) via universal instantiation. Finally, from W_4(d_1) we can infer exists y_2: W_4(y_2) via existential generalization. Therefore, the answer is yes.