I begin in the name of Allah, the most kind and merciful:

Summarized Answer

Scholars, past and present, have differed over this issue, so it should not be turned into a matter of dispute. It appears to me that Muslims should pray both the ʿĪd and Friday prayer. However, whoever decides to follow the other opinion must pray Ẓuhr and will not incur any sin, since this is a legitimate opinion based on proper reasoning and evidence as well. Mosques must organize the Friday prayer for those people who would like to attend.

Reason for the Difference of Opinion

Pieces of Evidence

A: The verse of the Qur’an stipulates that Friday prayer is an obligation: “You who believe: When the Prayer is called for Friday, hasten toward the remembrance of Allah and leave your business. That is better for you, if only you knew.”[1]

B: There is a report that the third khalīfah, Uthmān ibn ʿAffān, gave permission for some people to skip the prayer: “…then I witnessed the ʿĪd with Uthmān ibn ʿAffān, and that was on Friday. He prayed before the sermon [khutbah], then gave a speech and said: ‘People. This is a day where two ʿĪds have fallen on the same day. So whoever from amongst the people of the outskirts^[2] of Madinah wants to wait for the Friday Prayer, they may; and whoever wants to return [home], I have given them permission.”[3]

C: There are reports that the Prophet allowed people to skip the ʿĪd prayer.

• Zayd ibn Arqam reported that the Prophet performed the ʿĪd prayers early in the day but then offered an exemption for Friday prayer and said, “Whoever wants to may pray it.”[4]
• “Two ʿĪds were on the same day during the time of Ibn al-Zubayr [a Companion]. He delayed people from coming out until the daylight had spread. When he came out and gave a sermon, he made it long. Then he descended and prayed but the people did not pray the Friday Prayer on that day. This was then mentioned to Ibn ʿAbbās who said: ‘He has acted according to the Sunnah [the way of the Prophet].’”[5]
• Abū Hurayrah reported that the Prophet said, “Two ʿĪds have synchronized together on this day, so whoever prefers, it may suffice for Friday prayer. We will soon gather.”[6]

D: There is a report that the Prophet himself performed the Friday prayer on ʿĪd day: “The Prophet used to read surah al-Aʿlā and al-Ghāshiyah in the two ʿĪd Prayers and the Friday Prayer. When the day of ʿĪd and Friday would come together on the same day he would still read both of them in both prayers.”[7]

First Opinion

Only people living in isolated areas [who are normally exempted from attending the Friday prayer] coming from out of town to attend the ʿĪd prayer are exempted from the Friday Prayer. This is the opinion of Abū Ḥanīfah[8], Mālik[9], and al-Shāfiʿī[10].

Reasoning Behind the First Opinion

• Verse A cannot be overridden by any report which indicates something different unless it is of the highest authenticity. It must also be reported by several different people because this is not something that would only be heard/observed by one or two people only.
• There doesn’t seem to be any rational reason why one obligation should be dropped due to another being performed. This is similar to the way people must still pray Ẓuhr after praying ʿĪd.[11]
• Report B indicates that the leader of the Muslims exempted only a specific group of people [who normally don’t need to pray the Friday prayer because they don’t live in a city] and none of the Companions objected to his decision. This implies they understood that it was in line with the practice of the Prophet.
• Evidence C may be general in wording but should be understood as being confined to a specific group of people based on the other evidence.
• Much of evidence C is of doubtful authenticity.
• Report D indicates that the Prophet himself prayed it and he obviously had other people with him.

Second Opinion

Whoever performed the ʿĪd prayer is exempted from the Friday prayer[12], but must still pray Ẓuhr. This is the opinion of Aḥmad[13]. It is also the opinion of later scholars such as Ibn Taymiyyah[14], al-Shawkānī, Ibn Bāz[15], and Sayyid Sābiq.

Reasoning Behind the Second Opinion

• Evidence C is sufficiently authentic to prove that the Prophet made an exception to the rule in order to make life easier for the Muslims.
• The sermon for Friday prayer is an addition to the prayer of Ẓuhr. Since one set of sermons was already heard, there is no need for another set later in the day.
• Friday prayer is a type of ʿĪd and there is no need for two of them in one day. When two acts of worship of the same genre combine together, one of them drops, the way wuḍū’ is not needed when taking a bath [ghusl].[16]

Conclusion

There is clearly a legitimate difference of opinion due to both the clarity and authenticity of the two reports in question.