The Mishnah discusses a debate between Beis Shammai and Beis Hillel regarding the permissibility of carrying on Yom Tov. Rav Yitzchok bar Avdimi explained that their argument hinges on the principle of ‘mitoch’. Rabbah raised a difficulty with this explanation which was resolved by Rav Yosef.
Rabbi Yochanan is also of the opinion that the debate hinges on the principle of ‘mitoch’*:A learner was learning in the presence of Rabbi Yochanan: One who cooks a gid hanashe in milk, on Yom Tov, and eats it, is liable for five sets of malkos. 1. For cooking a gid [which is not something meant to be eaten, he thus cooked on Yom Tov for nothing]. 2. For eating gid hanashe. 3. For cooking meat and milk. 4. For eating meat and milk. 5. For kindling a fire on Yom Tov.Rabbi Yochanan: Go teach this outside [you’re not saying it correctly], kindling a fire and cooking [1 & 5] is not a teaching. If indeed this was taught, it was taught by Beis Shammai who do not hold of ‘mitoch’. According to Beis Hillel, however, kindling a fire is permitted on Yom by the principle of ‘mitoch - since it is allowed for ochel nefesh, it is allowed for other purposes as well’**.
__________________________________* Tosfos: This principle is only applied to a hotza’ah which is at least somewhat necessary for Yom Tov, otherwise it’s a Torah prohibition of carrying [for this reason, while it’s permitted to take a machzor to shul, one may not carry it back home, unless he fears it will go missing, in which case we can apply the principle of ‘the end was permitted because of its beginning’ (052)]. A mitzvah necessity such as Milah is also considered a Yom Tov necessity. This is the opinion of Rabeinu Chananel, but Tosfos argues that one may carry out a child even for mere enjoyment and not just because of a mitzvah, since enjoyment can also be considered a ‘Yom Tov necessity’.** Tosfos: Since he desires to eat it on Yom Tov, it is considered a Yom Tov necessity for himself. 
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