Incredible doctrine of a bygone age (4,6)

I believe the answer is:

past belief

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'incredible doctrine' is the definition.
I don't know anything about this answer so I can't judge whether this works.

'bygone age' is the wordplay.
'bygone' becomes 'past' (similar in meaning).
'age' becomes 'belief' (I can't explain this - if you can you should give a lot more credence to this answer).
'past'+'belief'='PAST BELIEF'

'of a' is the link.

Can you help me to learn more?

I've seen this clue in The Irish Independent, The Irish News and The Globe and Mail.

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