A bible going out for primate in Asia (5,6)

I believe the answer is:

agile gibbon

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'primate in asia' is the definition.
I don't know anything about this answer so I cannot judge whether this works.

'a bible going out' is the wordplay.
'out' is an anagram indicator (out can mean wrong or inaccurate).
'bible'+'going'='biblegoing'
'biblegoing' with letters rearranged gives 'gilegibbon'.
'a'+'gilegibbon'='AGILE GIBBON'

'for' is the link.

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I've seen this clue in The Times.

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