Before us is Mishnah 2 in the sixteenth chapter of Yevamos, which deals with two yevamos, each testifying that her own husband has died, and with the question of how far a woman's credibility extends: to herself alone, or to her counterpart as well.
The first case - two yevamos testifying to the deaths of their husbands:
"Shtei yevamos, zo omeres meis ba'ali v'zo omeres meis ba'ali" - the case is of two women married to two brothers, Reuven and Shimon, whose husbands traveled overseas. Each of them returns and testifies that her husband has died.
"Zo asurah mipnei ba'alah shel zo v'zo asurah mipnei ba'alah shel zo" - Reuven's wife is forbidden on account of Shimon, who is her yavam, and Shimon's wife is forbidden on account of Reuven, since they are not believed with regard to one another concerning the death of the other's yavam.
The underlying principle lies in the scope of a woman's credibility:
For herself - she is believed: A woman is believed when she says that her husband died, and she may even act upon it and remarry on the strength of this testimony.
For her counterpart - she is not believed: A yevamah is not believed to testify to the death of the yavam. She is believed regarding her husband's death for her own needs alone, but as for the implications of this for her counterpart, she is not believed.
"L'zo eidim ul'zo ein eidim":
One of them has witnesses to her husband's death, and the other has nothing but her own word. The result seems to run against intuition: "Es sheyeish lah eidim asurah" - even though her husband's death was established by witnesses, whose testimony is effective for everyone, the death of her yavam rests solely on the word of her counterpart, who is not believed with regard to her, and we do not know that no yavam is alive. In contrast, "v'es she'ein lah eidim muteres" - regarding her own husband she is believed by her own right, and regarding the death of her brother-in-law, who is her yavam, there are witnesses.
"L'zo banim ul'zo ein banim":
"Es sheyeish lah banim muteres" - for she has no obligation of yibum at all. "V'es she'ein lah banim asurah" - because the obligation of yibum rests upon her, and although we believe her that her husband died, we do not believe that her yavam died.
"Nisyabmu u'meisu ha'yevamin":
Until now we have dealt with a case where we do not know whether the yavam died. Now the case is that each of them has an additional yavam, a brother who is not overseas but is here. Since we believe that their husbands died and they have no children, the obligation of yibum applies, and they enter into yibum with the brother who is present. Afterward the yevamin die as well, and they are widowed from them.
"Adayin asuros l'hinasei" - perhaps there is yet another yavam, namely the brother who remained overseas, and so a new obligation of yibum applies to them (all of this assuming they have no children from the yavam either). True, each of them has already remarried on the strength of her husband's death, and if her husband indeed died, there is no yavam here at all, but we do not believe her regarding the death of the yavam as it affects her counterpart, only regarding her own husband, as his wife, and therefore they are still forbidden.
The view of Rabbi Elazar:
Rabbi Elazar disagrees with this last point and says: "Ho'il v'hutru la'yevamin, hutru l'chol adam" - since they already acted and married through yibum on the strength of their husbands' deaths, for there was a yavam present, once they were permitted to the yevamin they are permitted to any man, and we assume that the yavam died and there is no obligation of yibum, since no yavam exists.
In summary: In this Mishnah we learned the limit of a yevamah's credibility - she is believed regarding her husband's death for her own needs, and she is not believed regarding the yavam's death as it affects her counterpart. From here come the four rulings: both are forbidden, each on account of the other's husband; the one who has witnesses is forbidden and the one who has no witnesses is permitted; the one who has children is permitted and the one who has no children is forbidden; and even after they entered into yibum and the yevamin died, they are forbidden to remarry, while Rabbi Elazar permits it, since once they were permitted to the yevamin they are permitted to any man.