In the Name of Allah, the Benevolent, the Merciful
According to the vast majority of Islamic scholarship–including the Hanafis, Malikis, and Shafi`is–it remains obligatory to perform the Friday prayers when Eid falls on a Friday. [Nawawi, al-Majmu` Sharh al-Muhadhdhab; Ibn Abidin, Radd al-Muhtar] The Hanbalis allow skipping the Friday prayer in such circumstances, however. (Thus, one shouldn’t criticize nor condemn someone who takes that opinion–while striving to follow the former position.)
This is because the clear Divine Command to hasten to Friday prayer isn’t lifted by non-definitive, non-detailed individual texts that can be understood in conformity with the general ruling established by the clear verse:
“O believers, when the call to prayer is given on Friday then hasten to the remembrance of Allah and leave your business. This is better for you, if you but knew.” [Qur’an, 62.9]
There is a hadith in which Sayyiduna `Uthman gave permission for those who came from outside Medina to skip the Friday prayer. [Related by Imam Malik in his Muwatta’ and by Imam Bukhari in his Sahih] This permission is understood to return to the general rule that villagers and travellers are not obligated to perform the Friday prayer. [Muhammad ibn al-Hasan, Muwatta’; Nawawi, al-Majmu` Sharh al-Muhadhdhab; Ibn Abidin, Radd al-Muhtar]
Eid prayer fell on Friday during the time of the Prophet (Allah bless him and give him peace), and it is established that he performed both the Eid and Friday prayers.
Imam Muslim and others relate from al-Nu`man ibn Bashir (Allah be pleased with him) that, “The Messenger of Allah (peace and blessings be upon him) used to recite Sabbih Isma Rabbika’l A`la and Hal Ataka Hadhithu’l Ghashiya in both Friday and Eid prayers. And if Eid and Friday occurred on the same day, he (peace and blessings be upon him) would recited these in both.” This hadith makes it clear that the Prophet (Allah bless him and give him peace) used to perform the Friday prayer if Eid fell on a Friday.
And Allah alone gives success.