If the two Eids (al-Fiṭr/al-Aḍḥā and Jumuʿah) fall on the same day, it is not obligatory to attend Jumuʿah for those who have attended the Eid prayer, they have the option of praying Ẓuhr prayer instead. If a male Muslim missed the Eid prayer for whatever reason, Jumuʿah prayer still remains obligatory upon him to attend. Jumuʿah prayer is not obligatory on women.
Muʿāwiyah asked Zayd ibn Arqam, ‘Were you ever with the Messenger of Allāh (SAW) when two Eids happened on the same day?’ He replied, ‘Yes.’ He said, ‘What did he do?’ He said, ‘He offered the Eid prayer, then he granted a concession allowing people to miss Jumuʿah prayer, and he said, “Whoever wishes to pray (Jumuʿah), let him do so.”‘ (Abū Dāwūd)
Allāh’s Messenger (SAW) said, “Two Eids have come together on this day of yours, so whoever wishes, it [the Eid prayer] will suffice for Jumuʿah, but we will pray Jumuʿah.” (Abū Dāwūd)
Ibn ʿUmar said that the two Eids came together on the same day at the time of Allāh’s Messenger (SAW). He led the Eid prayer and then said, “Whoever wishes to come to Jumuʿah may come and whoever wishes not to do so may stay away.” (Ibn Mājah)
ʿAbdullāh asked his father, Imām Aḥmad about a person leaving one of the two Eids when both fall on the same day and he replied, ‘There is no problem in doing so.’ Ibn Qudāmah said, ‘If Eid and Jumuʿah fall on the same day, it is no longer obligatory to attend Jumuʿah for someone who has attended Eid.’