The Mishnah (4:5) teaches that the oldest brother should be the one to perform the mitzvah of yibum. If he does not want to, then we approach each of the other brothers in age order. If no one wants to, then we return to the oldest brother and assert that it is his mitzvah, and either yibum or chalitzah must be performed.
The next Mishnah (4:6) however discuss other cases where the brothers who declined taking action, might have a different claim. That might say that they should wait for the young minor brother to grow up and allow him to perform yibum. Alternatively, they might claim that they should wait for their older brother to return from overseas (or recover). In both cases we reject those claims. We say to the oldest brother present that he must perform either yibum or chalitzah. Why?
The Bartenura explains that even if the brother that is overseas is the oldest brother, with whom it would be the best way to perform the mitzvah, we still do not delay. He explains that this is based on the general principle that we simply do not delay the performance of mitzvah. Why?
The Melechet Shlomo cites the Terumat HaDeshen that notes that the Mishnah selects the example of waiting for a brother to come from overseas rather than from place to place within Israel. He explains that it is specifically the cases of one coming from overseas and waiting for the young brother to grow up, since the young brother might die, or the older brother might not come and then the mitzvah will not be fulfilled. He understands that we would have to wait for the older brother to come from place to place within Israel. In other words, the concern with delaying the mitzvah is when there is a concern that it could compromise its fulfilment.
The Terumat HaDeshen (35) cited by the Melechet Shlomo was responding to question whether the practice of waiting until motzei Shabbat to perform kiddush levanah is correct. The Terumat HaDeshen responds by differentiating between how many days remain until the middle of the month after motzei Shabbat. He cites the Ohr Zarua that it is best to recite it on motzei Shabbat when one is dressed in their finest clothing. If however, there are only a few days left till the middle of the month after motzei Shabbat then there is a risk that there will be a few cloudy nights in a row and they will miss out on the mitzvah.
The Terumat HaDeshen cites our Mishnah as the proof. Since there is a debate whether the yibum of the youngest is preferable to the chalitzah of the oldest, both cases of waiting for the youngest or oldest brother are included in the Mishnah. This is to stress that despite the fact that if we wait, the mitzvah might be able to be performed in the best possible way, if there a risk that the mitzvah will not be performed, the mitzvah must be performed without delay.
The Mishnah Berurah (Biur Halacha 426, s.v. be'motzei) cites the Bach however that there is an issue with delay in performing a mitzvah -- period. To be clear, this issue is not that the delay might compromise the performance, but rather delaying performing a mitzvah itself is problematic. Consequently, the Bach is against delaying performing kiddush ha'chodesh and one should perform the mitzvah immediately at the earliest time, after three days have past (from the molad).
We find a similar position in the Magen Avraham (25:2). The Rama rules that if one has tefillin and but not tzitzit, he does not need to wait for tzitzit, but rather put tefillin on immediately. The Magen Avraham explains that this is because we do not delay in performing a mitzvah even at the expense of performing it later in a better manner. He cites the Yalkut that explains that performing a mitzvah in its time is itself desirable -- chavivah mitzvah be'shaata.
The Nemukei Yosef (12b, Rif, s.v. Alecha) also explains that we do not delay performing a mitzvah, however he also adds we do not want to keep the yavama like aguna, preventing her from remarrying. In other words, we do not want to delay chalitzah since it is detrimental to the yavama.
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