Gemara

5b - Question: Is a woman's first Bi'ah permitted on Shabbos?

Is (virginal) blood stored up (so it is permitted)?

Or, is it absorbed (in the walls of the womb, and it comes due to a wound made by Bi'ah Rishonah, so it is forbidden on Shabbos?

6a: Rav permits, even though he rules like R. Yehudah (who forbids Davar she'Eino Miskaven). If the blood is absorbed, he holds that the wound is destructive.

7a: Rav Yehudah permitted Bi'ah Rishonah on Yom Tov.

Rav Papi: He permits even on Shabbos. The case brought before him just happened to be on Yom Tov.

Rav Papa: He permits only Yom Tov, but not Shabbos.

Question (Rav Papi): Do you hold that Mitoch (since) wounding is permitted when needed (for food, i.e. Shechitah), it is also permitted when not needed? If so, it should be permitted to burn incense on Yom Tov (since burning is allowed for food, it should be allowed even when not needed)!

Answer (Rav Papa): "What every person will eat" - the Torah allows only what is common to all (i.e. most) people.

Question (Rav Acha Brai d'Rava): If so, Shechitah of deer should be forbidden on Yom Tov, since not all people eat deer!

Answer (Rav Ashi): Something that all people desire is allowed. All desire deer (although not everyone can afford it).

Beitzah 12a - Mishnah - Beis Shamai: One may not be Motzi (take out) a child, Lulav or Sefer Torah to Reshus ha'Rabim (on Yom Tov);

Beis Hillel permit.

A reciter of Beraisos: If one slaughtered an Olas Nedavah on Yom Tov, he is not lashed.

R. Yitzchak bar Avodimi: This is like Beis Shamai. Beis Hillel say that Mitoch that Hotza'ah is permitted for a need (of Yom Tov), it is also permitted without a need, and similarly regarding Shechitah!

Question (Rabah): Perhaps Beis Shamai and Beis Hillel do not argue about Mitoch, rather, about whether or not Hotza'ah is forbidden on Yom Tov!

Answer (Rav Yosef): If so, they would have argued about rocks!

The following shows that also R. Yochanan holds that they argue about Mitoch.

A reciter of Beraisos: If one cooked Gid ha'Nasheh in milk on Yom Tov and ate it, he receives five sets of lashes. He is lashed for cooking a Gid (that may not be eaten) on Yom Tov, for eating a Gid, for cooking meat and milk, for eating meat and milk, and for burning on Yom Tov.

R. Yochanan: Do not teach this in the Beis Medrash! No one is Mechayev for burning or cooking on Yom Tov!

If someone is Mechayev, it is Beis Shamai, who do not say Mitoch regarding Hotza'ah, and likewise regarding burning (and cooking);

Beis Hillel say Mitoch regarding Hotza'ah, and likewise regarding burning (and cooking).

Rishonim

Rif and Rosh (Beitzah 1:18): Beis Shamai do not say 'Mitoch that Hotza'ah is permitted for a need, it is also permitted without a need.' Beis Hillel say Mitoch. 'For a need' means for a need of food. 'Without a need' refers to Hotza'ah of a child, Lulav or Sefer Torah.

Rambam (Hilchos Yom Tov 1:4): If on Yom Tov one did any Melachah for which one is liable on Shabbos, if it was not for the sake of eating he is lashed. The only exceptions are Hotza'ah and burning. Mitoch that Hotza'ah is permitted for a need, it is also permitted without a need. Therefore, one may take a child, Sefer Torah, key or similar things from Reshus to Reshus. Similarly, one may burn even if not needed for eating. All other Melachos are permitted only for the sake of eating, such as Shechitah, baking and kneading. What is not needed for eating is forbidden, such as writing, weaving and building.

Ra'avad: Burning not for eating refers to heating water to wash one's legs. The Yerushalmi says 'do not permit nor forbid a lamp of Avtalah (not for illumination, e.g. it is for honor).'

Magid Mishnah: In the Yerushalmi, R. Yochanan said not to permit nor forbid a lamp of Avtalah. The Rambam follows our Gemara, in which R. Yochanan himself says that Mitoch totally permits burning.

Questions (Lechem Mishneh): The Magid Mishnah understands that the Rambam exempts any Melachah done for a need, even writing or weaving. This cannot be. Also, he says that one is exempt for other Melachos (normally) for eating, e.g. Shechitah and cooking, if not done for eating. He does not give a reason. This implies that this is more obvious than the exemption for Hotza'ah and burning. The Gemara disagrees. It says that one who is Mechayev for unnecessary Shechitah holds like Beis Shamai. According to the Magid Mishnah, even Beis Shamai only argue about Hotza'ah and burning, but we have no source that they argue about Shechitah and cooking! The Magid Mishnah says that "Lo Seva'aru Esh b'Chol Moshvoseichem b'Yom ha'Shabbos" permits burning on Yom Tov. But R. Noson uses this verse to Mechalek (to be Mechayev for each Melachah by itself), and R. Yosi uses it to teach that burning on Shabbos is only a Lav! Also, the Gemara says that it is permitted due to Mitoch! Also, the Rambam exempts one who cooked for Nochrim because of Ho'il (since Yisraelim guests might come, perhaps he cooks for a need), not due to Mitoch! The Magid Mishnah says that the Rambam totally exempts Hotza'ah. But he forbids cooking and Shechitah for Nochrim!

Answers (Lechem Mishneh): Indeed, the Gemara holds that in truth even Beis Shamai Mechayev only for Hotza'ah, but not for Shechitah. It is better to expound directly (about Shabbos, that burning is Mechalek or only a Lav) than to infer about Yom Tov. However, "b'Yom ha'Shabbos" is extra, so it also teaches about Yom Tov. Beis Hillel permit Hotza'ah on Yom Tov from "V'Lo Sotzi'u Masa mi'Bateichem b'Yom ha'Shabbos". We infer that a Masa (load) is permitted on Yom Tov, even though it is not needed. Therefore, Mitoch is not needed for Hotza'ah. It is needed for burning, so we do not say that "Lo Seva'aru" permits on Yom Tov only for food. Perhaps we do not expound "V'Lo Sotzi'u...b'Yom ha'Shabbos" to exclude Yom Tov, for the Isur is learned from another verse, "Va'Ya'aviru Kol ba'Machaneh". However, the other questions are difficult.

Rosh (1:18): R. Chananel says that the Mishnah permits taking a child to circumcise him, a Sefer Torah to read and a Lulav to fulfill the Mitzvah. Mitoch permits only things that are somewhat needed, e.g. for a Mitzvah. If we would not say so, one who cooked on Yom Tov for the sake of Chol would not be lashed. The Gemara forbids burning (Pasul) Kodshim on Yom Tov, and forbids burning incense because not everyone does so. All the more so, something not needed at all on the day is forbidden. Olas Nedavah is also considered a need of the day, for it is improper that your table is full (of meat of Shelamim) and your Master's table (the Mizbe'ach) is empty. Even though it is forbidden to eat Gid ha'Nasheh, since one gets Simchas Yom Tov through eating it, it is considered a need of the day. Likewise, Bi'as Mitzvah is considered Simchas Yom Tov. One may not extinguish to permit Bi'ah because it is like Machshirin (preparations).

R. Tam permits to take a child for other needs, e.g. if one cannot leave the child alone at home. One may take Seforim to the Beis ha'Keneses, and even return them. This is so people will take them to begin with.

Rosh (2:22): The Yerushalmi neither permits nor forbids a lamp of Avtalah. Lamps lit in the Beis ha'Keneses are not of Avtalah, for they are for a Mitzvah and Simchas Yom Tov. Mitoch permits them, just like a child, Lulav or Sefer Torah.

Poskim

Shulchan Aruch (518:1): Mitoch that Hotza'ah is permitted (on Yom Tov) for needs of eating, it is also permitted without a need, e.g. a child, Sefer Torah, Lulav or Kelim.

Beis Yosef (495 DH v'Chen): The Magid Mishnah explains that the Rambam exempts Hotza'ah and burning even without a need. If any other Melachah was done without a need, one is liable. Since Hotza'ah is done to food itself, we exempt even when it is done to other things. The Yerushalmi permits burning on Yom Tov from "Lo Seva'aru..."

Rema: This is if he needs them somewhat, or if he fears lest they be stolen or another loss.

Kaf ha'Chayim (6): Some say that the Shulchan Aruch permits without any need at all. Others say that the Shulchan Aruch does not discuss this, for the Beis Yosef was undecided. The Rema explains that the Shulchan Aruch's intent is to forbid.

Question (Magen Avraham 1): "Lo Seva'aru..." implies that one may burn on Yom Tov. But the Gemara (Pesachim 5a) forbids burning Kodshim on Yom Tov due to "Lo Sa'aseh Chol Melachah", i.e. we do not say Mitoch!

Answer: The Gemara in Pesachim is like R. Noson, who says that "Lo Seva'aru" is Mechalek (obligates for each Melachah by itself; burning is a Melachah. Our Gemara holds like R. Yosi, that "Lo Seva'aru" teaches that burning is only a Lav, and it is permitted on Yom Tov due to Mitoch.)

Rejection (Magen Avraham): The Gemara there said that we infer (1) that "Lo Seva'aru" is Mechalek, and (2) that we do not say Mitoch. According to this answer, these are one matter! Also, it is difficult to say that Beis Hillel and Beis Shamai argue like R. Noson and R. Yosi.

Mishnah Berurah (1): We say Mitoch also regarding burning, Shechitah, baking and cooking, for these Melachos pertain to Ochel Nefesh. There is a tradition that "Ach Asher Ye'achel" refers to every Melachah normally done for food, even when it is not done for food. However, it must be needed somewhat, just like Hotza'ah.

Shulchan Aruch (2): A Yisrael may not be Motzi anything for the sake of a Nochri.

Sha'ar ha'Tziyon (22): This is unlike the Shulchan Aruch said above, that Mitoch permits all Hotza'ah! Perhaps here he forbids mid'Rabanan. Some say that Hotza'ah for a Nochri is worse than for no need at all.