Digging books up, leaving nothing out (2,4)
I believe the answer is:
in toto
'leaving nothing out' is the definition.
('in toto' means in total or overall)
'digging books up' is the wordplay.
'digging' becomes 'into' (to dig something is to appreciate it or be into it).
'books' becomes 'OT' (Old Testament).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'ot' backwards is 'to'.
'into'+'to'='IN TOTO'
(Other definitions for in toto that I've seen before include "In entirety" , "overall" , "Altogether, completely" , "Wholly" , "Completely - entirely (Latin)" .)