Consider the following premises. exists x: P_4(x) <-> T(x). forall x: ~W_3(x). forall x: T(x) -> W_3(x). Can we infer forall x: P_4(x) from them? If possible, name the inference rules being used at each step.
 From forall x: T(x) -> W_3(x), forall x: ~W_3(x) we can infer forall x: ~T(x) via universal modus tollens. Finally, from exists x: P_4(x) <-> T(x), forall x: ~T(x) we can infer exists x: ~P_4(x) via existential biconditional elimination, which contradicts forall x: P_4(x). Therefore, the answer is no.