Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother in Islam, thanks for your question, which emanates from a God-fearing heart, since it shows your commitment to Prayer, the cornerstone of Islam. We commend your pursuit of Islamic counseling. Allah Almighty says, "The true believers are those who hear the advice and act upon the best of it." (Az-Zumar 39:18)
As for the point you raised in your question, it is subject to controversy among the Muslim scholars. However, we can say that it is better to pray both `Eid and Jumu`ah as long as this does not lead to hardship or inconvenience.
Responding to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states,
Jumu`ah is an essential symbol of Islam, which Muslims are obligated to establish, regardless of whether we have prayed `Eid prayer or not; `Eid is a Sunnah, and, therefore, its performance does not or cannot absolve us from establishing Jumu`ah.
However, while all schools of jurisprudence agree on the fact that we must establish Jumu`ah in spite of holding `Eid prayers, there is a difference of opinion among them on the issue, namely, whether everyone who has attended `Eid prayer needs to be present at the Jumu`ah as well. Both the Hanafi and Maliki Schools unanimously assert that praying `Eid does not lift the obligation from anyone of praying Jumu`ah. They ask: How could a fard (obligation) which has been mandated by the Qur’an be lifted by a Sunnah?
Both the Shafi`i and Hanbali schools, however, hold a different view. According to them, an exception is made for those who have performed `Eid prayer to allow them to skip Jumu`ah. They base their position on certain reports transmitted from the Prophet (peace and blessings be upon him) which indicate that he did allow people to skip Jumu`ah if they wished to do so after having attended `Eid prayer.
However, as explained by some scholars of the Shafi`i school, the above concession is especially granted for those who are coming to the city for `Eid prayer from far away suburbs or villages, in which case, it may prove to be a real inconvenience for them to make another trip for Jumu`ah.
When seen in this light, it is best to look at this concession as it really is: simply a concession granted to those who are otherwise burdened or inconvenienced from undertaking two separate journeys. It is, therefore, most reasonable to assert that one must not consider the above permission to skip Jumu`ah as a blanket permission for everyone.
As one scholar has rightly put it, "It is best to attend it." And even if certain individuals were to make use of this concession, as a community we are held accountable to establish the Jumu`ah.
The traditions that have been cited by scholars for granting the above concession are not at all of a status that can be considered as unquestionable. This fact has always remained a major point of contention in dividing the scholarly opinion on this issue.
One of the most popular ones cited in this respect is that reported by Abu Dawud on the authority of Abu Hurairah, which states that the Prophet (peace and blessings be upon him) said, "In this day two `Eids have fallen concurrently, so if anyone wishes to skip the Jumu`ah he may do so, but we are holding the Jumu`ah anyway." The hadith belongs to the category of mursal traditions (hadith with a broken chain of narrators), and, therefore, does not belong to the category of a well attested tradition of the first degree. However, after having examined the various isnads(chains of transmissions) and similar reports on the issue, Imam Ibn Hajar has concluded that the above report can be considered as being worthy of citation.
In conclusion, the special permission to skip the Jumu`ah granted to those who have prayed `Eid on Friday is simply an exception given to individuals, and not a general directive issued to the community to cancel the Jumu`ah prayer.
Jumu`ah must be held and the imam is held accountable for it. In such contentious issues, as has been rightly pointed out by one scholar, the safest route to follow is to make use of this allowance only when someone is facing a genuine inconvenience or hardship.
We must never lose sight of the fact that Jumu`ah is an important symbol of Islam, and, hence we are obliged to treat it with due reverence.
May Allah show us truth as truth and help us to follow it, and show error as error and help us to shun it, amen.
Allah Almighty knows best.