Can you cast a spell with flashback from exile if it was exiled by a different effect?

Flashback (CR 702.34) grants a card the ability to be cast from your graveyard for an alternative cost. The ability specifically reads that you may cast the card from your graveyard. Once you do, flashback then arranges for the card to be exiled instead of going anywhere else when it leaves the stack — that exile is a consequence of using flashback, not a trigger that grants flashback.

If a card ends up in exile through some other effect — say, an opponent's Leyline of the Void, or a removal spell like Surgical Extraction sending it there — the card is simply in exile with no special permission to be cast. Flashback's permission clause only applies while the card is in the graveyard (CR 702.34a). Being in exile is a completely different zone, and no ability carries over just because the card has flashback printed on it.

This is reinforced by CR 400.1, which notes that objects in different zones are treated independently, and CR 702.34a, which explicitly states the flashback ability functions from the graveyard.

Example: Your opponent casts Relic of Progenitus and exiles your Momentary Blink directly from your graveyard. Even though Momentary Blink has flashback, it is now simply an exiled card with no way to be cast — the flashback window (casting from graveyard) was never used.

Bottom line: Flashback is a graveyard-based ability. Exile reached by any route other than the flashback ability itself does not grant any casting permission.