Friday prayer and Eid prayer occurring on the same day
If the day of Eid occurs on Friday, then Friday prayer is no longer an obligation upon those who performed the Eid prayer.
On the authority of Zaid ibn Arqam (May Allah be pleased with him), "The Prophet (Peace be upon him) prayed the eid prayer and then he gave an exemption concerning the Friday prayer, saying: 'Whoever wishes to pray it may pray it'." (Recorded by the five and Al-Haakim; Ibn Khuzaimah calls it authentic)
On the authority of Abu Hurairah (May Allah be please with him), the Prophet (Peace be upon him) said, "Two festivals have occurred together on this day of yours. For whosoever desires, this will suffice for his Friday prayer, but we are going to perform Friday prayer." (Recorded by Abu Dawud)
It is preferred for the imaam to perform the Friday prayer so anyone who wishes to perform it may do so as well as those who were not able to attend the Eid prayer. The Prophet (Peace be upon him) said: "We are going to perform the prayer Friday." (Recorded by Abu Dawud)
According to Hanbali, the Dhuhr prayer is obligatory upon anyone who does not attend the Friday prayer because he has performed the Eid prayer.