Rather, must it not be that the baraita is equating them with regard to their immersion, in that both one who has relations with a menstruating woman and one impure with impurity imparted by a corpse immerse during the day?

The Gemara rejects that conclusion. No; actually, the baraita is referring to the duration of their impurity. Although there is nothing novel in that equation, it was necessary for the tanna to teach the latter clause of the baraita: However, in one sense, the case of one who is impure because he had relations with a menstruating woman is more severe than one impure with impurity imparted by a corpse. One who is impure because he had relations with a menstruating woman transmits impurity through lying on a bed or sitting on a chair, even if he never came into direct contact with the chair or the bed. He renders the bed or chair impure with a mild form of impurity. He confers upon them first-degree ritual impurity status, sufficient only to render foods and liquids impure. One impure with impurity imparted by a corpse transmits impurity by means of direct contact.

Come and hear an additional proof that Rabbi Ḥiyya taught: For the zav and the zava and the leper and the female leper, and one who had relations with a menstruating woman, and one who is ritually impure with impurity imparted by a corpse, their immersion is during the day. For a menstruating woman and a woman after childbirth, their immersion is at night. The Gemara concludes that this baraita is indeed a conclusive refutation of the opinion of Rav Shimi from Neharde’a, who said that one who is impure because he had relations with a menstruating woman immerses at night.

With regard to the sequestering of the High Priest, the Gemara asks: And before you remove him from the potential of impurity of his house, remove him from the potential of the more severe impurity imparted by a corpse. The Sages should have instituted an ordinance prohibiting visitors to the High Priest lest one die while in his chamber and render him impure. Rav Taḥlifa, father of Rav Huna bar Taḥlifa, said in the name of Rava: That is to say that impurity imparted by a corpse is permitted in cases involving the public. In cases where the public is involved, impurity imparted by a corpse does not prevent the Temple service from being conducted. Since the service of the High Priest on Yom Kippur is a service involving communal offerings, impurity imparted by a corpse does not invalidate the service.

Ravina said: Even if you say that impurity imparted by a corpse is merely overridden in public and not completely permitted, the idea that the High Priest is not removed from the potential of impurity imparted by a corpse can be understood. Impurity imparted by a corpse is uncommon. The likelihood that one visiting the High Priest will suddenly die is minimal. In contrast, impurity of his house is common, as uncertainty with regard to his wife’s status as a menstruating woman could arise at any moment.

There is an amoraic dispute with regard to the effect of impurity imparted by a corpse on the conduct of the Temple service. It was stated with regard to impurity imparted by a corpse that Rav Naḥman said: It is permitted in cases involving the public; e.g., when a majority of the Jewish people is impure, the service of a ritually pure priest is not preferable to that of an impure priest. The Temple service proceeds as though there was no impurity at all. And Rav Sheshet said: Impurity imparted by a corpse is merely overridden in cases involving the public, and service performed by a ritually pure priest is preferable.

The Gemara restricts the scope of the dispute. In a case where there are both ritually impure and pure priests in that patrilineal family tasked with serving in the Temple on that day, everyone, even Rav Naḥman, agrees that the pure priests serve and the impure priests do not serve. When they disagree, it is with regard to a case where the entire patrilineal family is impure. Is it necessary to seek out and bring pure priests from a different patrilineal family belonging to the same priestly watch, who are tasked with serving in the Temple on a different day that week?

Rav Naḥman said: The prohibition against performing the Temple service in a state of impurity imparted by a corpse is permitted in cases involving the public, and we do not need to seek out other priests. Since the Torah permitted the performance of the Temple service by priests impure with impurity imparted by a corpse, it is completely permitted and it is as though the service is performed in purity. Rav Sheshet said: The prohibition of impurity imparted by a corpse is overridden in cases involving the public, and wherever possible we seek out ritually pure priests.

Some say that the dispute is slightly different: Even in a case where there are both ritually pure and ritually impure priests in that patrilineal family, Rav Naḥman disagreed with Rav Sheshet and said that the priests serve even when they are impure,