Consider the following premises. forall x_23: P_23(x_23) <-> Q_23(x_23). ~q_3. q_3 <-> P_23(a_19). Can we infer Q_23(a_19) from them? If possible, name the inference rules being used at each step.
 From forall x_23: P_23(x_23) <-> Q_23(x_23) we can infer forall x_23: Q_23(x_23) -> P_23(x_23) via universal biconditional elimination. From forall x_23: Q_23(x_23) -> P_23(x_23) we can infer Q_23(a_19) -> P_23(a_19) via universal instantiation. From q_3 <-> P_23(a_19), ~q_3 we can infer ~P_23(a_19) via biconditional elimination. Finally, from Q_23(a_19) -> P_23(a_19), ~P_23(a_19) we can infer ~Q_23(a_19) via modus tollens, which contradicts Q_23(a_19). Therefore, the answer is no.