KEDUSHAS BECHOR DOES NOT APPLY TO KOHANIM AND LEVIYIM
(After Chet ha'Egel, the Bechoros were disqualified from Avodas ha'Mishkan. The Leviyim (including Kohanim) were taken in place of 22,000 of them. There were 273 surplus Bechoros. Each had to redeem himself for five Shekalim. The animals of the Leviyim were in place of those of the Bechoros. This was in the second year in the Midbar. We call this "at that time.")
(Mishnah): A Kal va'Chomer exempts (Pitrei Chamorim of) Kohanim and Leviyim (from Kedushas Bechor). If Kohanim and Leviyim exempted Yisraelim in the Midbar, all the more so they (their animals) are exempt!
(Gemara) Question: People (Kohanim and Leviyim) exempted people (Yisraelim), and (their) animals exempted animals (of Yisraelim) - "Kach Es ha'Leviyim Tachas Kol Bechor bi'Vnei Yisrael v'Es Behemas ha'Leviyim Tachas Behemtam." (However, what is the source that people (Leviyim) exempt (their own) animals?)
Answer #1 (Abaye): The Mishnah means, since the Seiyin (this is the plural of 'Seh') of Kohanim and Leviyim exempted Pitrei Chamorim of Yisraelim in the Midbar, all the more so they themselves are exempt!
Objection #1 (Rava): The Mishnah says that Kohanim and Leviyim (themselves) exempted (Yisraelim)!
Objection #2 (Rava): If so, even Tahor Behemos of Kohanim and Leviyim should be exempt!
(Mishnah): Kohanim and Leviyim are exempt only from Kedushas Bechor of Peter Chamor and people (but not from Bechor Behemah Tehorah).
Answer #2 (Rava): The Mishnah means that since Kohanim and Leviyim exempted (people, i.e.) Yisraelim in the Midbar, all the more so they themselves are exempt.
Question: What is the source that their animals exempt those of Yisrael?
Answer: "Ach Pado Sifdeh Es Bechor ha'Adam v'Es Bechor ha'Behemah ha'Teme'ah Tifdeh" teaches that Kedushas Bechor of Behemah Teme'ah (i.e. donkeys) applies only to animals of people obligated to redeem themselves (i.e. Yisraelim).
Question #1 (against Abaye - Rav Safra): You say that the Kal va'Chomer applies also to animals. It is valid only for a Levi (in the Midbar) who had a Seh (at that time, to redeem Behemos Yisrael), but (Shitah - descendants of) a Levi in the Midbar who did not have a Seh should have (Rashi - had) to redeem his own Pitrei Chamorim! (Questions 2, 3 and 4 are against Abaye and Rava.)
Question #2 (Rav Safra): Only Leviyim at least one month old (at that time) exempted Yisraelim. We should say that a Levi less than one month did not exempt himself! (We do not find that such Leviyim needed to be redeemed.)
Question #3 (Rav Safra): Only male Leviyim exempted Yisraelim. Children of a Bas Levi should not be exempt!
(Rav Ada bar Ahavah): The Bechor of a Bas Levi need not be redeemed.
Answer (Mar brei d'Rav Yosef): Kedushas Bechor depends on "Peter Rechem," therefore a Bas Levi exempts her son just like a Levi does. (Rashi - this answer depends on finding an answer to Question #2, that exemption is not limited to Leviyim who actually exempted Yisraelim; Tosfos - this answer stands by itself.)
Question #4: Aharon was not counted (he did not exempt a Yisrael). His Pitrei Chamorim should not have been exempt!
(Beraisa): There are are dots over Aharon's name (in the Parshah of counting the Leviyim) to show that he was not counted.
Answer (to Questions 1, 2 and 4): "Ha'Leviyim" equates all the Leviyim to each other. (All of their Pitrei Chamorim are exempt).
Question: What is the source that also (Aharon and other) Kohanim are equated?
Answer: We learn from R. Yehoshua ben Levi.
(R. Yehoshua ben Levi): In 24 verses, Kohanim are called Leviyim, e.g. "veha'Kohanim ha'Leviyim Bnei Tzadok."
Question: What is the source that they are exempt for all generations?
Answer: "V'Hayu Li ha'Leviyim" teaches that their law will not change.
A SEH REDEEMED PITREI CHAMORIM IN THE MIDBAR
Question: What is the source that a Yisrael's Peter Chamor was exempted by a Levi's Seh (at that time in the Midbar)?
Answer #1 (Rav Chisda): For all generations, the Torah discusses redeeming (people) with money and (Pitrei Chamorim) with a Seh;
Just like money was also used to redeem at that time, also a Seh.
Objection: We cannot learn from money. It has more ability to redeem. It can redeem Hekdesh and Ma'aser Sheni!
Answer #2: It says "Ach Pado Sifdeh Es Bechor ha'Adam v'Es Bechor ha'Behemah ha'Teme'ah Tifdeh";
Just like the same thing (i.e. money) used to redeem firstborn sons for all generations was used at that time, the same thing (a Seh) used to redeem a Peter Chamor for all generations was used at that time.
(R. Chanina): One Seh of a Levi redeemed (at that time) many Pitrei Chamorim of Yisraelim.
Support #1 (Abaye): This is surely true, for the Torah counts how many more Bechoros there were than Leviyim, but not how many Pitrei Chamorei Yisrael there were in excess of Seiyin of Leviyim.
Objection: Perhaps it does not count animals because there was no excess. The Leviyim had more!
Answer: "U'Mikneh Rav Haysah li'Vnei Reuven uli'Vnei Gad." ("Mikneh" includes donkeys).
Rejection: Still, perhaps the Seiyin (that were not Bechoros) of the Leviyim exceeded the number of Pitrei Chamorim of Yisrael!
Support #2: "V'Es Behemas ha'Leviyim Tachas Behemtam" - one Behemah of a Levi exempted many Behemos of Yisrael.
Question: "Behemah" can refer to many animals - "Behemah Rabah"!
Answer: If each Levite Seh exempted only one animal, it should have said "Behemah Tachas Behemah," or "Behemtam Tachas Behemtam";
Rather, it uses the singular for the Leviyim and the plural for Yisrael, to teach like R. Chanina.
Support #3 (Rava - Mishnah): The same Seh can redeem many times. (Rashi - after buying it back from the Kohen, the Yisrael can redeem another Peter Chamor with it. R. Tam - a Yisrael can redeem many Safek Pitrei Chamorim with one Seh. He need not give it to a Kohen);
R. Chanina explains that the same Seh can redeem many times, because one Seh of a Levi in the Midbar redeemed many Pitrei Chamorim of Yisraelim.
KEDUSHAS BECHOR IN THE MIDBAR
Version #1 (R. Yochanan): Bechoros born in the Midbar became Kodesh.
(Reish Lakish): They did not become Kodesh.
R. Yochanan learns from "Kadesh Li Chol Bechor";
Reish Lakish says right after "v'Hayah Chi Yevi'acha (Hash-m El Eretz ha'Kena'ani)" it says "v'Ha'avarta (Chol Peter Rechem la'Shem)," implying that Kedushas Bechor will not apply until entering Eretz Yisrael.
Question (R. Yochanan - Mishnah): Before the Mishkan was erected, Bamos were permitted, and firstborns did the Avodah;
After the Mishkan was erected, Bamos were forbidden, and Kohanim did the Avodah;
Answer (Reish Lakish): The Bechoros who left Mitzrayim did the Avodah.
Support: Bechoros born in the Midbar were too young to do Avodah (when the Mishkan was erected). The Mishkan was set up less than a year after leaving Miztrayim!
Question: The answer is obvious. Why did R. Yochanan ask the question?!
Answer: He holds that the Kedushah of Bechoros that were already Kadosh would have lapsed in the Midbar if newly born Bechoros were not becoming Kadosh.
Reish Lakish holds that previous Bechoros retained their Kedushah, but new Bechoros did not become Kadosh.
Version #1A - Question (R. Yochanan - Beraisa): On the day that the Mishkan was set up, Yisrael offered Nedarim and Nedavos, Chata'os and Ashamos, and Bechoros and Ma'aseros.
Answer: They offered Bechoros that left Mitzrayim.
Support (for Reish Lakish): We infer that Bechoros were offered only on the day that the Mishkan was set up, but Bechoros born in the Midbar did not become Kodesh!
Version #1B - Question (Reish Lakish - Beraisa): On the day that the Mishkan was set up, Yisrael offered Nedarim and Nedavos Chata'os and Ashamos, and Bechoros and Ma'aseros.
Inference: Bechoros were offered only on the day the Mishkan was set up, but Bechoros born in the Midbar did not become Kodesh!
Answer: They offered from that day and onwards.
Question: If so, what Chidush does the Beraisa teach?
Answer: Obligatory Korbanos were not offered before this, for they may not be offered on a Bamah.
Question (Beraisa): Bechoros became Kodesh in three places - in Mitzrayim, in the Midbar, and when Bnei Yisrael entered Eretz Yisrael;
In Mitzrayim it says "Kadesh Li Chol Bechor";
In the Midbar it says "Ki Li Chol Bechor... "
Upon entering Eretz Yisrael, it says "v'Hayah Chi Yevi'acha... v'Ha'avarta (Chol Peter Rechem)."
Answer (Rav Nachman bar Yitzchak): It means that in three places Bnei Yisrael were commanded about Kedushas Bechor, but Bechoros did not become Kodesh.
Question: They did become Kodesh in Mitzrayim!
Answer: He means that in some of these three places (i.e. in the Midbar), Bechoros did not become Kodesh, but in other places they did.
Objection: (Rav Papa): They did become Kodesh in the Midbar - "Pekod Kol Bechor Zachar bi'Vnei Yisrael" (this applied "mi'Ben Chodesh va'Ma'alah," even though all births in the last year were in the Midbar)!
Version #2 (R. Yochanan): (Starting in Mitzrayim,) Bechoros became Kodesh. This never ceased;
(Reish Lakish): Kidush Bechoros ceased (at the time of the redemption in the Midbar. It resumed upon entering Eretz Yisrael.)