P.I. Alec's problem birth brings important mail (7,8)

I believe the answer is:

special delivery

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

'important mail' is the definition.
'special delivery' can be an answer for 'mail' (special delivery is a kind of mail). I'm not sure about the 'important' bit.

'p i alec's problem birth' is the wordplay.
'problem' indicates anagramming the letters.
'birth' becomes 'delivery' (delivery is a kind of birth).
'p'+'i'+'alecs'='pialecs'
'pialecs' anagrammed gives 'special'.
'special'+'delivery'='SPECIAL DELIVERY'

'brings' is the link.

Can you help me to learn more?

(Other definitions for special delivery that I've seen before include "What a courier may have brought" , "that a spinner hopes to produce?" .)

I've seen this clue in the Sydney Morning Herald.

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