Consider the following premises. P(a_21) -> S(a_21). w -> S(a_21). forall z_2: S(z_2) -> P(z_2). ~q_2. P(a_21) <-> q_2. Can we infer w from them? If possible, name the inference rules being used at each step.
 From forall z_2: S(z_2) -> P(z_2) we can infer S(a_21) -> P(a_21) via universal instantiation. From S(a_21) -> P(a_21), P(a_21) -> S(a_21) we can infer S(a_21) <-> P(a_21) via biconditional introduction. From P(a_21) <-> q_2, ~q_2 we can infer ~P(a_21) via biconditional elimination. From S(a_21) <-> P(a_21), ~P(a_21) we can infer ~S(a_21) via biconditional elimination. Finally, from w -> S(a_21), ~S(a_21) we can infer ~w via modus tollens, which contradicts w. Therefore, the answer is no.