Aboo Dawood (no. 1070), an-Nisaa‘ee (3/194), Ibn Maajah (no. 1310), Ibn Khuzaymah (no. 1474) ad-Daarimee (no. 1620) Ahmad (4/372) transmit from Iyaas Ibn Abee Ramlah as-Shaamee who said: ‘‘I witnessed Mu’aawiyah Ibn Abee Sufyaan - Have you witnessed the coinciding of two ’Eeds on the same day with the Prophet (sallallaahu ’alayhi wa sallam)?  He replied: ‘Yes.’  So he said: ‘What did he do?’  He replied: ‘He (sallallaahu ’alayhi wa sallam) prayed ’Eed and then gave permission not to pray Jumu’ah, and said: ‘He amongst you who desires to pray, let them pray.’’ [104] In this area, Aboo Hurairah and others report from the Prophet (sallallaahu ’alayhi wa sallam).  The Companions also act according to this.

’Abdur-Razzaaq reports in his Musannaf (3/305), and Ibn Abee Shaybah in his Musannaf (2/187) with a authentic chain of narration from ’Alee Ibn Abee Taalib that two ’Eeds coincided on the same day, so he said: ‘‘He who wishes to pray Jumu’ah then let him do so and he who wishes to sit then let him do so.’’ And in Saheehul-Bukhaaree is a similar narration from ’Uthmaan (radiyallaahu ’anhu).

In Sunan Aboo Dawood (no. 1072), and in the Musannaf of ’Abdur-Razzaaq (no. 5725) with an authentic chain of narration from Ibn Zubayr that he said: ‘‘Two ’Eeds coincided on the same day and so they were both joined together by making them one.  He prayed on the day of Jumu’ah two rak’ahs in the morning and he did not add to them until he prayed the ’Asr Prayer ...’’  Ash-Shawkanee said in Naylul-Awtaar (3/348) after this narration: ‘‘It is apparent that he did not pray Dhuhr Prayer.’’   It also shows that if for any (permissible) reason, the Jumu’ah Prayer is cancelled it is not obligatory for the person for whom it has be cancelled to pray the Dhuhr Prayer and ’Ataa‘ held this opinion.

Thus it is apparent that those who hold this opinion are those who say that the Jumu’ah Prayer is  the origin.  You are well aware of what He obligated over his servants on the day of Jumu’ah, and that is the Jumu’ah Prayer. Thus making the Dhuhr Prayer obligatory upon a person who leaves it  i.e. Jumu’ah, either due to a permissible excuse or otherwise is in need of evidence and there is no evidence which is worthy of being adhered to from that which I know.

What to do when `Eed coincides with Friday
By The Permanent Committee for Islaamic Research and Fataawa

Question: ...he gave a fatwa that the Jumu'ah and Zuhr prayer are "dropped" (not obligatory) if one attended the `Eed prayer whether it is the imaam or (any of) the congregation,...

Response: If the `Eed coincides with Friday, then attendance to the Jumu'ah prayer for anyone who attended the `Eed prayer is "dropped" (not obligatory) except for the imaam. For certainly he is not relieved of this obligation except if the people do not gather for the Jumu'ah (prayer).

And with Allaah lies all the success, and may Allaah send prayers and salutations upon our Prophet Muhammad (sal-Allaahu `alayhe wa sallam) and his family and his companions.

The Permanent Committee for Islaamic Research and Fataawa, comprising -
Head: Shaykh 'Abdul 'Azeez ibn Abdullaah ibn Baaz;
Deputy Head: Shaykh 'Abdur-Razzaaq 'Afeefee;
Member: Shaykh 'Abdullaah Ibn Qu'ood
Fataawa al-Lajnah ad-Daa.imah lil-Buhooth al-'Ilmiyyah wal-Iftaa. - Volume 8, Page 179, Fatwa No.2140

What to do when `Eed coincides with Friday
By Shaykh Ibn Baaz

Question: If any of the two `Eeds fall on a Friday (Jumu'ah), do you pray the Jumu'ah (prayer) with its khutbah on that day or not?

Response: That which is legislated for the Muslims if the `Eed coincides with Friday is to pray the `Eed prayer and the Jumu'ah prayer in the masaajid (plural of masjid) where the Jumu'ah prayer is (normally) held. And it is permissible for one who attended the `Eed prayer to leave off the Jumu'ah and instead perform the Zuhr prayer.

Shaykh Ibn Baaz
Majmoo' Fataawa Samaahatu ash-Shaykh Ibn Baaz - Volume 4, Page 504, Fatwa No. 556