Consider the following premises. forall x_3: P_3(x_3) -> W(x_3). forall x_3: P_3(x_3) or ~W(x_3). forall x_3: ~P_3(x_3). Can we infer exists x_3: W(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_3(x_3) or ~W(x_3) we can infer forall x_3: W(x_3) -> P_3(x_3) via universal rewriting. From forall x_3: W(x_3) -> P_3(x_3), forall x_3: P_3(x_3) -> W(x_3) we can infer forall x_3: W(x_3) <-> P_3(x_3) via universal biconditional introduction. Finally, from forall x_3: W(x_3) <-> P_3(x_3), forall x_3: ~P_3(x_3) we can infer forall x_3: ~W(x_3) via universal biconditional elimination, which contradicts exists x_3: W(x_3). Therefore, the answer is no.