Consider the following premises. exists y: Q(y). forall y: Q_4(y) or S_3(y). forall y: Q_4(y) <-> R(y). forall y: ~R(y). Can we infer forall y: ~(Q(y) and S_3(y)) from them? If possible, name the inference rules being used at each step.
 From forall y: Q_4(y) <-> R(y), forall y: ~R(y) we can infer forall y: ~Q_4(y) via universal biconditional elimination. From forall y: Q_4(y) or S_3(y), forall y: ~Q_4(y) we can infer forall y: S_3(y) via universal disjunctive syllogism. Finally, from exists y: Q(y), forall y: S_3(y) we can infer exists y: Q(y) and S_3(y) via existential conjunction, which contradicts forall y: ~(Q(y) and S_3(y)). Therefore, the answer is no.