Consider the following premises. forall y: S_4(y) -> P_4(y). forall y: P_4(y) <-> Q(y). forall y: Q(y) -> S_4(y). Can we infer P_4(a) -> Q(a) from them? If possible, name the inference rules being used at each step.
 From forall y: Q(y) -> S_4(y), forall y: S_4(y) -> P_4(y) we can infer forall y: Q(y) -> P_4(y) via universal transitivity. From forall y: P_4(y) <-> Q(y) we can infer forall y: P_4(y) -> Q(y) via universal biconditional elimination. From forall y: Q(y) -> P_4(y), forall y: P_4(y) -> Q(y) we can infer forall y: Q(y) <-> P_4(y) via universal biconditional introduction. From forall y: Q(y) <-> P_4(y) we can infer Q(a) <-> P_4(a) via universal instantiation. Finally, from Q(a) <-> P_4(a) we can infer P_4(a) -> Q(a) via biconditional elimination. Therefore, the answer is yes.