Answered by Shaykh Hamza Karamali, SunniPath Academy Teacher
If Jumah coincides with Eid, is it still obligatory to perform Jumuah? Please elaborate as much as possible.
[From shafii mailing list]
In the Name of Allah, Most Merciful and Compassionate
Shaykh Muhammad Zahid al-Kawthari al-Hanafi al-Ashari (Allah have mercy on him), one of the greatest scholars of recent times (d. 1371 AH), full of ghayra (jealousy) for the way of Ahl al-Sunna, encyclopedic in his knowledge of the entire spectrum of the Islamic sciences, dealt with the issue of Eid and the Friday prayer in detail in an article that is published as part of his invaluable collection of Maqalat. Almost everything that follows is taken from his article on pgs. 249-257 of Maqalat al-Kawthari.
Imam al-Shafii, Imam Abu Hanifa, Imam Malik, and all their immediate students (ashab) concurred that if Eid falls on a Friday, the villagers and bedouins who come from out of town to attend the Eid prayer are exempted from performing the Friday prayer if they leave after the performance of the Eid prayer, but city-dwellers must still perform the Friday prayer. The Hanbalis disagreed and said that the performance of the Eid prayer exempts everyoneboth villagers and city-dwellersfrom the obligation of performing the Friday prayer.
The evidence that the three schools use for their position is the hadith related by Imam Malik in his Muwatta and Imam Bukhari in his Sahih from the mawla of Ibn Azhar who said:
I witnessed Eid with Uthman b. Affan (Allah be pleased with him). He prayed and then left and then gave a sermon, saying: Verily, two Eids have occurred simultaneously for you today, so whoever from among the people of Aliyah  wants to wait for the Friday prayer may wait and whoever wants to leave may leave, for I have given him permission.
Imam Muhammad b. al-Hasan al-Shaybani (Allah have mercy on him) said after relating this hadith in his compilation of the Muwatta:
We adopt all of this. Uthman only excused the people of Aliyah from performing the Friday prayer because they werent from the people of the city. This is the position of Abu Hanifah.
Abul-Walid al-Baji mentioned several people in his commentary on the Muwatta who related from Imam Malik similar to what is related from Imam Abu Hanifa.
Imam al-Shafii said in al-Umm:
If Eid al-Fitr falls on a Friday, the Imam prays Eid when the prayer becomes permissible and then gives permission for the non-city-dwellers who attend the Eid prayer to leave if they wish back to their homes and not return for the Friday prayer. It is preferable for them to remain until performing the Friday prayer orif ableto return after leaving to perform the Friday prayer, although if they dont, there is no harm, in-shaAllah. This [h: meaning leaving the Friday prayer] is not permissible for any of the city-dwellers without an excuse [h: that exempts them from the obligation of performing the Friday prayer].
There are hadiths related by Abu Dawud that seem to point to the position of the Hanbalis, but Shaykh al-Kawthari masterfully shows how all these hadiths are either too weak to be used as evidence because of problems with the narrators or are interpreted in the light of the hadith related by Bukhari (mentioned above). Those interested can refer to pgs. 253-257 of his Maqalat.
The Hanbalis have their own arguments for their position, of course, and it is valid for a muqallid of the Hanbali school to follow their position.
And Allah knows best.
 Aliyah is what is above the Najd until Tihamah, miles away from Madinah.