Eduyos, Chapter 7, Mishnah 6. In this Mishnah, Rabbi Yehoshua and Rabbi Papyas, who already appeared in the previous Mishnah, return once again, testifying together to a single piece of testimony: regarding the status of an offspring born to an animal that was consecrated as a shelamim (peace offering). The animal was consecrated, and before it was offered it gave birth to a young animal - and the status of that offspring is the subject of the discussion.
On a Torah level everyone agrees that the offspring of a shelamim is itself a shelamim, and it is to be treated exactly as one treats the parent animal. Rabbi Eliezer, however, holds that on a rabbinic level we do not treat the offspring as a shelamim; rather, the lamb, kid or calf is left to die. His reasoning: a rabbinic decree, out of concern that people might delay bringing their shelamim until the animal gives birth to an offspring.
Such a delay has negative consequences:
Nullifying the positive commandment to bring the offering at the first opportunity, when a person comes to visit the Beis HaMikdash.
Transgressing the prohibition of 'bal te'acher' - the prohibition against delaying the bringing of an offering, which one violates after three consecutive festivals have passed.
The Bartenura points out the concern before us: if a person is given an incentive not to offer his shelamim, he will say to himself: if I leave my shelamim unoffered, it will produce a whole army of offspring and I will be able to bring a thousand offerings. But in the meantime he is liable to lose track of the animal and transgress the prohibition of working with or shearing consecrated animals. For this reason, according to this view, we do not want shelamim wandering around unoffered, and therefore a negative incentive was established: even the offspring of a shelamim is not brought as a shelamim, but is left to die, like the law of one who is placed in the kipah. This is the dispute before us.
The wording of the Mishnah:
"He'id Rabbi Yehoshua v'Rabbi Papyas al v'lad shel shelamim" - Rabbi Yehoshua and Rabbi Papyas testified concerning the offspring of a shelamim. The 'v'lad' is the offspring: the calf born to a cow designated as a shelamim, the kid born to a goat designated as a shelamim, or the lamb born to a ewe designated as a shelamim. "Yikrav shelamim" - it is offered as a shelamim in the Beis HaMikdash, just like any other shelamim.
And why was their testimony needed? Because "Rabbi Eliezer omer: v'lad shelamim lo yikrav shelamim" - Rabbi Eliezer says: the offspring of a shelamim is not offered as a shelamim. His position is that we do not offer the offspring the way we offer the parent, because of the concerns mentioned above. This is a rabbinic law, designed to prevent people from leaving their shelamim unoffered, lest mishaps occur. "VaChachamim omrim: yikrav" - and the Sages say: it is offered. The halachah was decided in accordance with Rabbi Yehoshua and Rabbi Papyas, that the offspring is offered in the very same way.
Rabbi Papyas's testimony:
Until now the question was discussed, as it were, on a theoretical level: what the tradition of these men was. Now Rabbi Papyas adds a precedent from his own personal experience, which follows his position: "Me'id ani shehaysah lanu parah zivchei shelamim" - I testify that we had a cow that was a shelamim offering. It was a cow brought as a shelamim, and after it was consecrated, before it was slaughtered, it gave birth to a calf, and the calf too had the status of a shelamim. "Va'achalnuha b'Pesach" - and we ate it on Pesach; we ate the original shelamim at the time of Pesach, as shalmei simchah. "V'achalnu v'ladah b'chag" - and we ate its offspring on the festival; the calf, its offspring, we ate as a shelamim on the festival that followed.
The straightforward reading here is that the festival Rabbi Papyas refers to is Shavuos, since, as mentioned, there is a positive commandment to bring the offering at the first opportunity, and we assume this was the Shavuos that immediately followed. Even though 'the festival' without qualification usually refers to Sukkos, in this case it apparently means Shavuos. Either way, the main point is that Rabbi Papyas has firsthand experience of treating the offspring of a shelamim as a shelamim.
In summary: the halachah was decided in accordance with the Sages and with the testimony of Rabbi Yehoshua and Rabbi Papyas - the offspring of a shelamim is considered a shelamim and is offered like any other shelamim, and not in accordance with Rabbi Eliezer's position.