Everybody's first cheers? (3,3,4)

I believe the answer is:

all the best

'cheers?' is the definition.
(I've seen this before)

'everybody's first' is the wordplay.
'everybody' becomes 'all' (I've seen this before).
'first' becomes 'thebest'.
'all'+'thebest'='ALL THE BEST'

(Other definitions for all the best that I've seen before include "A greeting" , "every member of the elite?" , "Pick of the bunch" , "needing good luck!" , "flower, every bit of it?" .)

I've seen this clue in The Guardian.

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