Does Eid Prayer substitute for Fajr Prayer, and does Fajr Prayer substitute for Eid Prayer?

The Eid prayer is a confirmed Sunnah (definition) (a highly recommended practice). Neither the Fajr prayer (whether its obligatory or Sunnah part) can substitute for it, nor does the Eid prayer suffice for Fajr. Each has its own time and specific ruling. If Eid and Friday coincide, the majority of jurists hold that the Eid prayer does not suffice for the Friday prayer. However, the Hanbalis believe that the Friday prayer is waived for those who perform the Eid prayer in this situation, and they pray Dhuhr instead. They cite the Hadith (definition) of Abu Hurairah: "Indeed, two Eids have coincided on your day; so, whoever wishes, it suffices him from Friday prayer, but we are gathering."