HOTZA'AH ON YOM TOV
(Beis Shamai): One may not carry out to Reshus ha'Rabim that which is not needed for Ochel Nefesh on Yom Tov (e.g. a child, a Lulav, a Sefer Torah).
(Beis Hillel): It is permitted.
MITOCH SHE'HUTRAH L'TZORECH
It was taught before R. Yitzchak b. Avdimi that one who slaughters an Olas Nedavah on Yom Tov incurs Malkos.
(R. Yitzchak b. Avdimi): The source of that teaching is Beis Shamai (who do not apply Mitoch to permit Melachah for non Ochel Nefesh purposes) and could not be Beis Hillel (who do make that extension).
Question (Rabah): Perhaps our Mishnah deals with the issue of whether Eruv and Hotza'ah are relevant to Yom Tov (Beis Shamai) or not (Beis Hillel, inferring from the Pasuk which speaks only of Shabbos), while both would agree not to apply Mitoch!?
Answer (R. Yosef): If Beis Hillel indeed hold that there is no Hotza'ah on Yom Tov, then they should have taught this regarding permitting stones (thus we may demonstrate that the argument is not over Hotza'ah itself, but over Hotza'ah she'Lo l'Tzorech)!?
We may demonstrate that R. Yochanan also holds that the Machlokes is over Mitoch (and that all agree to the existence of Isur Hotz'ah on Yom Tov).
It was taught before R. Yochanan that one who cooks Gid ha'Nasheh in Milk and eats it on Yom Tov incurs five sets of Malkos (as listed).
R. Yochanan rejected the inclusion of Hav'arah (which Mitoch should permit) unless the Mishnah were Beis Shamai (thus demonstrating his view that Beis Shamai and Beis Hillel argue over Mitoch).
MISHNAH: TRANSPORTING THE PERMITTED HAFRASHOS TO THE KOHEN ON YOM TOV
(Beis Shamai): One may not (give or) transport Chalah and Matanos (from an animal) to the Kohen on Yom Tov, regardless of whether the Hafrashah took place on or before Yom Tov.
(Beis Hillel): It is permitted to give and transport the Chalah and Matanos.
Beis Shamai adduces support from a Gezeirah Shavah (comparing Dinim d'Rabanan) linking Chalah and Matanos to Terumah (which, while taken before Yom Tov, may not be brought to the Kohen on Yom Tov).
Beis Hillel differentiate between Chalah and Matanos (which one is permitted to be Mafrish on Yom Tov and, by extension, to give and transport) and Terumah (which one is not permitted to be Mafrish on Yom Tov).
UNDERSTANDING THE MACHLOKES BETWEEN BEIS HILLEL AND BEIS SHAMAI
We initially assume that the Machlokes holds regardless of when the Hafrashos took place (before or on Yom Tov).
Question: If so (that Beis Shamai forbids giving even Matanos which were Mufrash on Yom Tov) then our Mishnah follows the view of Acherim (and does not follow the opinions of R. Yosi nor R. Yehudah)!?
(R. Yehudah): Even Beis Shamai agree that Hafrashos from Yom Tov may be given.
(R. Yosi): Chalah and Matanos may be given regardless of when the Hafrashah took place (and the argument is only over Terumah).
(Acherim): All agree that Terumah may not be given and the argument is only over the Matanos.
Answer (Rava): The Mishnah could be R. Yehudah, and the Mishnah speaks only of Matanos whose Hafrashah was on Yom Tov (but which were slaughtered before Yom Tov)!
Question: Under that assumption, then the Mishnah is the view of R. Yehudah, and not Acherim!?
Answer: It may be the view of Acherim and Beis Shamai only prohibit giving the Matanos of animals slaughtered before Yom Tov (but Matanos of animals slaughtered on Yom Tov would be permitted to be brought to the Kohen).
Question: Then the position of Acherim is identical with that of R. Yehudah!?
Answer: They would differ on whether it is permitted to add Matanos from an animal slaughtered before Yom Tov to the Matanos being brought from an animal slaughtered on Yom Tov (R. Yehudah) or not (Acherim).
(R. Yehudah citing Shmuel): The Halachah follows R. Yosi (that even Terumah may be given on Yom Tov).
When the question was brought before R. Yosef, he ruled in accordance with R. Yehudah citing Shmuel, like R. Yosi.
CRACKING OPEN SHELLS OF SEEDS ON YOM TOV
Question (the landlord): Is it permitted to crack open mustard seeds and eat them on Yom Tov?
Answer (Rava bar R. Chanan): He did not know the answer.
The question was brought to Rava, who ruled that one may roll ears of wheat, and crack kernels on Yom Tov.
Question: A Beraisa contradicts this!?
One who rolls ears of wheat on Erev Shabbos, on Shabbos may blow them (to remove the chaff) from hand to hand and eat the kernels, but not in a Kinun nor in a Tamchuy.
One who rolls ears of wheat before Yom Tov, may blow them on Yom Tov one by one and eat the kernels, even in a Kinun and a Tamchuy, but not in sifter or a sieve.
On Erev Yom Tov one may roll ears of wheat, but not on Yom Tov itself!
Answer: It is also permitted on Yom Tov (Erev Yom Tov is mentioned only to parallel the Reisha of the Beraisa which says Erev Shabbos).
Question: According to Rava (who permits opening the chaff on Yom Tov in order to eat the kernel inside), the statement of our Mishnah is inaccurate.
Our Mishnah teaches that Terumah cannot be separated on Yom Tov.
According to Rava, Terumah can be separated on Yom Tov, for otherwise one would not be permitted to eat the kernel from which he removes the chaff (which Rava permits)!