The fifth perek deals with the prohibition of oto v'et beno – the prohibition of slaughtering a mother animal and its offspring on the same day. When probing the scope of this prohibition, the Mishnah (5:3) teaches that if one fails in performing valid shechita or stabs or tears out the simanim of one of the animals, and performs shechita on the other, they are exempt – they have not violated the prohibition. That is because the prohibition only applies to slaughtering both on the same day, and in his case one of the animals died and was not slaughtered.
One detail worth addressing is that the Mishnah appears uncharacteristically wordy: "… he is exempt from the prohibition of ota v'et beno". Considering that the perek is dedicated to discussing this prohibition, it would have been enough had it written "he is exempt".
The Tifferet Yisrael explains that the Mishnah is teaching that after the first animals died by one of these means, the prohibition does not exist at all for the second animal. In other words, one is completely permitted to slaughter the second animal. In other words, had the Mishnah only written "he is exempt", one might have though that this is the case only after the fact, but ideally the second animal should not be slaughtered.
The Tosfot R' Akiva Eiger however present a different explanation. The fact that the Mishnah stressed from which prohibition he is exempt, implies that there are prohibitions that would be violated. By killing the animals in such manners, one violates the prohibitions of tzaar baalei chaim (causing pain to animals) and bal tashchit (waste).
The Tifferet Yaakov (114) however does not accept this answer, because the question of further issurim does not appear relevant to this perek. Whether or not those prohibitions apply is relevant to all animals and not just in the context of oto v'et beno, and may not apply in cases of need.
The Tifferet Yaakov therefore suggests that the closing of this Mishnah is important for the debate in the beginning. The Mishnah records the debate regarding a case where a valid shechita was performed to one of the animals, yet there was another reason that made the animal forbidden. For example, it was discovered that the animal was a treifa or it was slaughtered for idol worship. According to R' Shimon one is exempt if the second animal was slaughtered on the same days. Since the slaughter did not permit the animal for consumption, it is not considered shechita. The Chachamim (R' Meir) disagree and the prohibition of ota v'eta beno would apply to the second animal.1
The Tifferet Yaakov explains that the closing of the Mishnah appears to support those that maintain that the debate is only when the first animal slaughtered was, e.g. discovered to be a treifa. If however the first animal received a proper shechita, then even R' Shimon would agree that for the second animal, a shechita that does not permit the animal to be consumed is considered shechita for the purposes of oto v'et beno.2
The Tifferet Yaakov concludes that the words closing the Mishnah are necessary to teach that in the case where one of the animals was, e.g. stabbed even the Chachamim agree that there is no prohibition at all of oto v'et beno irrespective of the order in which the animals died. He adds that the Mishnah was ordered in this way, mentioning in the first case that R' Shimon said "he is exempt" so that the two "exemptions" can be contrasted. Since R' Shimon's statement is more concise, it is understood that he only exempts one in the case where the animal that was slaughtered first was discovered to treifa, but not if it was slaughtered second.
1 The Gemara records that the debate is based on exposition from pesukim.
2 See the Tifferet Yaakov who explains the distinction based on the how the R' Shimon's position is derived from pesukim.
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