We learned: Shabbos and Yom Tov that are contiguous: Rav: An egg born on one is forbidden on the following day. Rabbi Yochanan: It is permitted on the following day.
This is a debate among Tannaim: If it is laid on Shabbos, it can be eaten on Yom Tov [Sunday], or it was laid on Yom Tov [Friday], it may be eaten on Shabbos. Rabbi Eliezer: On the second day it is also a debate. Beis Shammai permits it, while Beis Hillel prohibit it.
The host of Rav Ada bar Ahavah had eggs that were laid on Yom Tov which fell on a Friday. He asked Rav Ada bar Ahavah whether he could broil the eggs on Yom Tov in order to eat them on Shabbos.
Rav Ada bar Ahavah: Do you think it is permissible because we follow Rabbi Yochanan over Rav? Even Rabbi Yochanan would only permit sipping the egg raw on Shabbos, but he wouldn’t allow handling it on Friday (an egg laid on Yom Tov is always muktzeh).
This is as we learned in a Beraisa (009): An egg that was laid on Shabbos or Yom Tov may not be handled, and it cannot be used to cover a jar or to support a wobbly bed.
The host of Rav Papa,
Some Say: Some man came to Rav Papa on Shabbos which was followed by a Yom Tov. He asked whether the eggs laid today would be permissible tomorrow on Yom Tov. Rav Papa asked him to return on the next day because [he had drunk wine at the meal (for Birkas Hamazon) and] Rav never set up an Amora to announce his Drasha (Rav would speak in a quiet voice and the Amora would announce it for everyone to hear) from one day of Yom Tov to the next [after he had eaten the Yom Tov meal] because of the wine.
On the next day, when he returned, Rav Papa said: Indeed now, I would have forgotten and said that the Halacha follows Rav Yochanan over Rav, but Rava said: In these three matters we follow Rav whether he’s stringent or lenient [the first thing being our case, that the egg is forbidden on the second day].
Rabbi Yochanan: Wood that fell from a tree directly into an oven on Shabbos, may not be used as firewood the following day on Yom Tov. Do not object to this based on my lenient ruling regarding an egg laid on Shabbos. Since an egg may be eaten on Shabbos, but we forbid it, it is permissible on the following day. But since firewood could not be used on Shabbos even if it is cut from the tree before Shabbos, the prohibition on Shabbos doesn’t mean anything and it is necessary to prohibit it on the following day as well.