Eiruvin, chapter five, mishnah five. This mishnah is the second half of the previous one, which brought the opinion of Rabbi Eliezer (and in another version: Rabbi Elazar), who added two items to the list of cases in which Beis Shammai are lenient and Beis Hillel are stringent. Before us is the second of the two, and it is also the last of the entire topic of disputes in which Beis Shammai are lenient and Beis Hillel are stringent.
The case:
The mishnah deals with marriage in a situation of yibum. There are four brothers: Reuven, Shimon, Levi and Yehudah. Two of the brothers, Reuven and Shimon, are married to two sisters, Rachel and Leah. Reuven and Shimon die without leaving children, and so the two widows, Rachel and Leah, fall to yibum or chalitzah before one of the remaining brothers, Levi or Yehudah, before they may marry anyone else.
Zikah - a bond resembling marriage:
As was explained earlier, there is a bond called zikah between the surviving brother and the widow of his deceased brother, and this bond looks and feels like marriage. Just as kiddushin - giving a ring with the declaration "Harei at mekudeshes li" - forbids the woman to every other man until the bond is resolved through marriage or through a get and divorce, so too a woman awaiting yibum stands in zikah to her yavam and cannot marry anyone until he resolves the bond through yibum or through chalitzah.
Therefore the Sages said, on a rabbinic level, that a woman bound to a man is to be treated in certain respects as though they were married. The main consequence is that some of her relatives become forbidden to him rabbinically, just as it is forbidden to marry a woman and her sister. This is the central prohibition with which our mishnah deals, called "achos zekukaso" - the rabbinic prohibition against marrying the sister of a woman to whom he has a zikah.
Applying the law to our case:
Rachel and Leah are the widows of Reuven and Shimon, and by Torah law the two remaining brothers, Levi and Yehudah, may perform yibum with one of them or with both, one for each brother. By rabbinic law, however, neither of them can marry either of the women. Let us focus on Levi: he cannot marry Rachel, since he already has a zikah to her sister Leah, and Rachel is therefore the sister of the woman bound to him; and for the very same reason he cannot marry Leah, since she is the sister of the woman bound to him on account of Rachel. And exactly the same applies to Yehudah, who is likewise barred from both women.
It emerges that, rabbinically, the only way forward is that the women release their zikah not through yibum but through chalitzah: one of the brothers, or both of them, gives chalitzah to each of the sisters, and thereby they are permitted to marry someone else, and there is no yibum here at all. This law is straightforward, and there is no dispute about it.
What is the law after the fact?
The question is what happens if Levi gave chalitzah to Rachel, and Leah is still waiting for yibum or chalitzah. Rabbinically, Yehudah, or Levi, should have given her chalitzah and not performed yibum with her. Yehudah, however, went ahead and performed yibum with Leah. This is a violation of a rabbinic law, since she was previously the sister of the woman bound to him; true, the zikah has now lapsed because Rachel has already received chalitzah, but they still should not have married. May they remain together, or are they compelled to divorce?
According to the Tanna Kamma (the anonymous voice of the mishnah) - they must divorce, and there is no doubt about it.
According to Rabbi Eliezer - this is a dispute between Beis Shammai and Beis Hillel: Beis Shammai are lenient and say that they may remain married, and Beis Hillel are stringent and say that they must divorce.
The halachah follows the anonymous Mishnah, that they are required to divorce, and this fits with the view of Beis Hillel according to Rabbi Eliezer. The main point here is the very existence of a dispute between Beis Hillel and Beis Shammai regarding the obligation to divorce, with Beis Shammai taking the lenient position that there is no such obligation.
Why are four brothers needed?
There is no essential difference between Levi and Yehudah, so the question arises: why did the Mishnah speak of four brothers and not just three? Reuven, Shimon and Levi; Reuven and Shimon are married to the sisters Rachel and Leah and they die, and Rachel and Leah fall to yibum before the remaining brother, Levi. He too cannot perform yibum with either of them, since each one is the sister of a woman bound to him. Why not say here too that if he performed chalitzah with one and yibum with the other, they may remain married?
The answer is that even according to Rabbi Eliezer, in this case Beis Shammai would be stringent and would not allow them to remain together. The reason: if Levi performed chalitzah with Rachel and then yibum with her sister Leah, the problem is not limited to the fact that Leah was previously the sister of a woman bound to him; there is an additional, separate rabbinic prohibition - the sister of the woman with whom he performed chalitzah. For chalitzah is regarded by the Sages as something very close to divorce, and effectively equivalent to it, and a man may not marry the sister of his former wife under any circumstances.
It emerges that even Beis Shammai agree that when there is only one brother and Levi first performed chalitzah with Rachel, Rachel is like a woman he released through chalitzah, comparable to a divorcee, and he can never marry her sister Leah during Rachel's lifetime. But when there are four brothers and two remain, Levi and Yehudah, and Levi performed chalitzah with Rachel, Yehudah has no problem of marrying the sister of the woman released through chalitzah when he marries Leah, and therefore if he did marry her, at least according to Beis Shammai they may remain married. The halachah, as stated, follows the agreed position, which on this matter reflects Beis Hillel as well according to Rabbi Eliezer, and therefore they may not remain married.
In the words of the Mishnah:
"Arba'ah achim" - four brothers: Reuven, Shimon, Levi and Yehudah.
"Shenayim mehem nas'u shtei achayos" - two of them married two sisters: Reuven and Shimon are married to Rachel and Leah.
"Meisu hanesu'im la'achayos" - Reuven and Shimon died, leaving their widows, Rachel and Leah, bound to the two remaining brothers, Levi and Yehudah, for chalitzah or yibum.
"Harei eilu cholatzos velo misyabmos" - they require chalitzah, and yibum is not an option by rabbinic law, since both brothers are barred from performing yibum with Rachel and Leah, as each of them is the sister of a woman bound to him.
"Ve'im kadmu vechansu - yotzi'u" - and if one of them performed yibum after the other woman had already undergone chalitzah, they are not left married but are compelled to divorce.
"Rabbi Eliezer omer mishum Beis Shammai: yekayemu" - the second brother, Yehudah, keeps the second woman after the fact, and they are not made to divorce, since she is no longer the sister of a woman bound to him and she never became the sister of a woman released through his chalitzah.
"U'Veis Hillel omrim: yotzi'u" - since they transgressed the rabbinic prohibition of marrying a woman who had been the sister of a woman bound to him, the marriage does not stand, and even after the fact they are compelled to divorce.
And the halachah follows Beis Hillel and the opinion of the first Tanna, that they are compelled to divorce.