Before us is Mishnah 9 in the third chapter of Maseches Yevamos. To appreciate its novelty, let us first set out the ordinary case: there were several brothers and one of them died, and another brother performed yibum with his widow, and afterward the one who performed yibum died as well - there is no problem here at all. The brother who performed yibum fulfilled the obligation of yibum, and that obligation lapsed. Now the woman falls to yibum solely by virtue of the second marriage, and this is a simple case of yibum in every respect.
Our Mishnah deals with an additional scenario, in which an obligation of yibum remains upon the woman from the first brother and from the second brother together.
The words of the Mishnah:
"Sheloshah achin nesu'in leshalosh nachriyos" - three brothers who are married to three women who are not related to one another.
"Umes echad mehem" - one of the brothers died, and his widow falls to yibum.
"Ve'asah bah sheni ma'amar, umes" - instead of the remaining brother actually performing yibum with her, the second brother made a ma'amar with her, which is a rabbinic form of betrothal, and afterward he too died.
"Harei elu cholzos velo misyabmos" - the two women, the widow of the first husband who received the ma'amar from the second, and the original wife of the second brother, perform chalitzah and do not undergo yibum.
According to this scenario the woman stands in a twilight zone between the two brothers: on the Torah level it appears that an obligation of yibum rests upon her from the first husband, and rabbinically also from the second husband by virtue of the ma'amar.
The reason for the ruling:
The verse states "yevamah yavo aleha," and from here the Sages derived that only a woman upon whom there is the bond of one yavam undergoes yibum, "velo she'aleha zikas shenei yevamin" - and not a woman who has the potential for yibum from two yevamin, from two different brothers. In our case the first brother died and afterward the second brother died, and each of them has a basis for a connection with her: the first was married to her, and the second made a ma'amar with her. For this reason two bonds rest upon her, and she is not obligated in yibum.
Chalitzah is still required, for as the Gemara says, this derivation is only an asmachta and not a full derivation, and the law is rabbinic. Since the ma'amar creates a connection on the rabbinic level alone, by Torah law the woman is obligated in a simple yibum from the first husband, and only from him is there a bond upon her. It is only because it appears as though two bonds rest upon her that she is not permitted to undergo yibum. The same is done with her co-wife, who is likewise not a full co-wife by Torah law and is considered her co-wife only on account of the ma'amar. Either way, for both of them only chalitzah is permitted and not yibum.
The view of Rabbi Shimon:
Rabbi Shimon has a different perspective on the ma'amar. In his opinion the ma'amar is not a connection resembling an acquisition, but rather a doubt: it is doubtful whether it takes effect or not. Either it takes effect completely, or it does not take effect at all. Accordingly, the case must be viewed in two ways: either the yevamah became the full wife of the second brother by virtue of the ma'amar, in which case there is no bond upon her at all from the first husband; or the ma'amar does not take effect at all, and she falls to yibum from the first husband alone, and the ma'amar is nothing.
On this basis Rabbi Shimon says: "Veyeyabem le'eizo sheyirtzeh, vechalitzah lasheniyah" - he performs yibum with whichever of the two women he wishes, and performs chalitzah with the second. For the woman who fell from the first husband either falls from him alone (if the ma'amar is ineffective), or she falls from the second husband. But it is impossible to perform yibum with both of them: if the ma'amar is effective, then both women are married to the second husband and fall from him alone, and one does not perform yibum with two women who fall from a single man.
An additional case - two brothers married to two sisters:
The Mishnah brings an additional case: "Shenei achim nesu'in lishtei achayos, umes echad" - two brothers who are married to two sisters, and one of the brothers died and his wife fell to yibum. At the moment she fell, the remaining brother cannot perform yibum with her, because she is his wife's sister. "Ve'achar kach mesah ishto shel sheni" - afterward the wife of the surviving brother died. Once the sister died, the widow is no longer his wife's sister, and seemingly the prohibition has been removed.
Nevertheless the Mishnah says: "Harei zu asurah alav le'olam", and the reason is - "Ho'il vene'esrah alav sha'ah achas". At the moment she first fell before him she was his wife's sister, who is an ervah, and there was no mitzvah of yibum with her at all; she was therefore a brother's wife with whom there is no mitzvah of yibum, and she is forbidden to him. And since she became forbidden to him at that moment, the prohibition remains in place even after the status of ervah lapsed with the death of his wife.
In summary: in this Mishnah we learned two rulings. The first: a woman who fell to yibum and with whom the second brother made a ma'amar and then died, has, as it were, the bond of two yevamin resting upon her, and therefore she and her co-wife perform chalitzah and do not undergo yibum, even though by Torah law her bond is from the first brother alone and the derivation is only an asmachta. Rabbi Shimon disagrees, since in his view the ma'amar is a doubt - either it takes effect completely or it does not take effect at all, and therefore he performs yibum with whichever he wishes and chalitzah with the second. The second ruling: two brothers married to two sisters, and the wife of the surviving brother died after his wife's sister had already fallen before him - the yevamah remains forbidden to him forever, since she became forbidden to him for one moment.