Consider the following premises. forall x_19: W_2(x_19) and Q_3(x_19). forall x_19: Q_19(x_19) or P(x_19). exists x_19: ~P_19(x_19). forall x_19: P_19(x_19) <-> Q_19(x_19). Can we infer exists x_19: P(x_19) and W_2(x_19) from them? If possible, name the inference rules being used at each step.
 From forall x_19: P_19(x_19) <-> Q_19(x_19), exists x_19: ~P_19(x_19) we can infer exists x_19: ~Q_19(x_19) via existential biconditional elimination. From forall x_19: Q_19(x_19) or P(x_19), exists x_19: ~Q_19(x_19) we can infer exists x_19: P(x_19) via existential disjunctive syllogism. From forall x_19: W_2(x_19) and Q_3(x_19) we can infer forall x_19: W_2(x_19) via universal simplification. Finally, from exists x_19: P(x_19), forall x_19: W_2(x_19) we can infer exists x_19: P(x_19) and W_2(x_19) via existential conjunction. Therefore, the answer is yes.