First lady always lacks Republican backing (3)

I believe the answer is:

eve

I'm a little stuck... Click here to teach me more about this clue!

'first lady' is the definition.
(I've seen this before)

I can't explain the rest of the clue.

(Other definitions for eve that I've seen before include "'All About - - ' (Bette Davis film)" , "One from Genesis" , "comes before Jan. 1st" , "See 13 Across" , "Night before an event" .)

I've seen this clue in The Times.

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