Gemara

Rav holds that Ba'alei Chayim are not Nidchim (if a living animal become Pasul for Hakravah, if the Pesul goes away the animal may be offered);

R. Yochanan holds that Ba'alei Chayim are Nidchim.

Question: What is Rav's source?

Answer #1: Rav learns from Mechusar Zeman. Even though it may not be offered now, when the time comes, it may be offered.

Rejection: Mechusar Zeman was never Kosher to be offered. He cannot learn from there to something that was Kosher and became Pasul.

Answer #2: Rav learns from a Mum that is Over (temporary). Even though it may not be offered now, when the Mum goes away it may be offered.

We know that it may be offered when the Mum goes away from "Ki Mashchasam Bahem Mum Bam" - they are Pesulim only when the Mum is Bam (in them, but not after it goes away).

R. Yochanan expounds "Bahem" - only a Ba'al Mum Over may be offered after the Pesul goes away, anything else (that was Kosher and later) Nidcheh may never be offered.

Rav expounds "Bahem" to teach (like R. Eliezer) that limbs of a Ba'al Mum are unacceptable by themselves, but they may be offered amidst a mixture:

Mishnah - R. Eliezer: If limbs of Olos Kesheros became mixed with limbs of a Ba'al Mum:

If one of the heads was offered, we offer all the heads. If the Kera'ayim of one of them were offered, we offer all the Kera'ayim;

Chachamim say, even if all of them except for one were offered, the remaining one is burned in Beis ha'Sereifah.

Avodah Zarah 73b - Rava: Since Chachamim forbade to offer the animals (they are Nidchim), if any were offered, they are Pesulim.

Question (Rav Huna bar Yehudah - Mishnah): If Chata'os ha'Of and Olos ha'Of became mixed together, even if one was mixed with 10,000 of the other, all must be left to die;

If a Kohen erred and offered all of them above (like Olas ha'Of), the Olos are Kesherim...(according to Rava, all should be Pesulim)!

Answer: Rava's law is like the opinion that Ba'alei Chayim are Nidchim. The Mishnah is like the opinion that Ba'alei Chayim are not Nidchim.

Question: All agree that a slaughtered animal can be Nidcheh, yet R. Eliezer says that (if one of a mixture was offered) it is Kosher!

Mishnah - R. Eliezer: If one of the heads was offered, we offer the rest of the heads.

Answer: R. Eliezer holds like Chanan ha'Mitzri (who holds that even slaughtered animals are not Nidchim):

Beraisa - Chanan ha'Mitzri: If Se'ir ha'Mishtale'ach died after Kabalas Dam of the Se'ir Chatas, we bring another goat to be the Se'ir ha'Mishtale'ach. The blood is still Kosher.

74a - Rav: If a ring of idolatry was mixed with 100 rings and one of them fell into the sea, all the rest are permitted. We assume that the forbidden one fell.

Question (Rava - Mishnah): Even if one (Chatas ha'Mesah or Shor ha'Niskal) was mixed with 10,000, all must die.

After the first dies, we do not assume that it was the forbidden animal!

Answer (Rav Nachman): Rav holds like R. Eliezer.

(Mishnah - R. Eliezer): If one of the heads was offered, we offer the rest of the heads.

Question: But R. Elazar said that R. Eliezer only permits to offer two at a time (since one of them is surely Kosher, we assume that the other is also), but not one at a time!

Answer (Rav Nachman): Indeed, Rav permits to sell the rest of the rings only two (or more) at a time.

Zevachim 77b: R. Eliezer learns from "Mum Bam" - it is forbidden to offer a Ba'al Mum by itself, but it may be offered among a mixture;

Chachamim expound "Mum Bam" - it is forbidden to offer while the Mum is there, if the Mum goes away, it is permitted;

R. Eliezer learns both laws from this, for the Torah wrote "Bam" instead of 'Bahem';

Chachamim say that this does not warrant learning a second law.

Rishonim

Rambam (Hilchos Pesulei ha'Mukdashim 6:1): If a Chatas ha'Mesah or a Shor ha'Niskal became mixed with even 10,000 Korbanos of any kind, all must die. This is because living beings are important, so they are not Batel. If someone offered, Hurtzah (it is acceptable), because Ba'alei Chayim are not Nidchim.

Kesef Mishneh: This is like the Gemara's answer for Rava, that Ba'alei Chayim are not Nidchim, therefore Hurtzah.

Lechem Mishneh: When they are in their place, this is considered Kavu'a and anything offered is Pasul. Hurtzah is only if they moved and then one was separated, for then we say that it came from the majority. The Rambam should have explained this!

Rebuttal #1 (Merumei Sadeh Zevachim 77b DH Yoser): The Lechem Mishneh understands that the Rambam says that if one of them was offered, Hurtzah. This is wrong. He means that if all of them were offered, Hurtzu.

Rebuttal #2 (Or Some'ach Ma'aseh ha'Korbanos 16:2 DH Hinei): The Rambam said that all must die because Ba'alei Chayim are not Batel. After Shechitah the Pasul one is Batel, just like nuts that are not Batel can become Batel if they are broken (Gitin 54b), therefore Hurtzah.

Question: The Rambam says that Ba'alei Chayim are not Nidchim. This implies that there is Dichuy after Shechitah. But the Rambam (2:23) says that if blood of Pesulim became mixed with blood of Kesherim (all of it must be spilled to the Amah,) and it was offered, Hurtzah!

Answer #1 (Merumei Sadeh ibid.): Really, the Rambam did not need to say that Ba'alei Chayim are not Nidchim to explain Hurtzah. He does not hold like Rava, who says that something which cannot be offered because it is mixed with Pesulim is Nidcheh. A Mishnah (Zevachim 79b) teaches that if Dam Pesulim became mixed with Kosher Dam and all was offered, Hurtzah. Rava must establish the Mishnah like Chanan ha'Mitzri, who says that Dichuy does not apply even after Shechitah. The Rambam need not do so.

Answer #2 (Keren Orah Zevachim 74a DH v'Yesh): There is Dichuy after Shechitah only if the Pesul became mixed after Shechitah. If it was mixed before Shechitah, we attribute the Pesul to the first one offered, and the rest are Kesherim.

Note: According to the Or Some'ach (ibid.), we can give an alternative answer. The Pasul blood is Batel because it is after Shechitah. Perhaps it is a Kenas not to offer it l'Chatchilah. The Rambam (Hilchos Ma'achalos Asuros 16:12) rules like R. Yosi who fines one who intentionally broke nuts so they will be Batel, that they are not Batel.

Rambam (ibid. 21): If limbs of Ba'alei Mumim became mixed with limbs of (Kosher) Kodshim, even one limb of a Ba'al Mum in 1000 Kosher limbs, all of them are burned where Pasul Kodshim are burned.

Ri Korkus: R. Eliezer holds that if a limb was offered, we assume that it was from the Ba'al Mum. This permits to offer the others. The Rambam rules like Chachamim.

Question: The Rambam (Hilchos Shegagos 3:8) rules like Rav, that Ba'alei Chayim are not Nidchim. The Gemara (Yoma 64a) says that Rav expounds like R. Eliezer!

Answer #1 (Lechem Mishneh Shegagos 3:8 DH Od): The Rambam holds like Tosfos (64a DH Ha) that Chachamim in the Mishnah (but not Chachamim of the Beraisa in Zevachim 74a) also expound "Bahem" to permit (mid'Oraisa) to offer limbs of a Ba'al Mum amidst a mixture. Chachamim forbid this mid'Rabanan.

Ritva (64a DH Ha): Rashi says that Rav holds like R. Eliezer. This is unreasonable. Why would Rav abandon Chachamim and hold like R. Eliezer?! Rather, Chachamim agree that the mixture is permitted mid'Oraisa.

Rebuttal (Sha'ar ha'Melech Korban Pesach 4:5 DH v'Ulam Ra'isi): The Ritva does not distinguish Chachamim of the Beraisa from Chachamim of the Mishnah. The Gemara (Zevachim 77b) explicitly says that Chachamim of the Beraisa do not expound "Bahem" to permit a mixture!

Answer #2 (Sha'ar ha'Melech, ibid.): Rav agrees that Dichuy applies after Shechitah.