6

Are those two interchangeable?

Can I say both: За умови and При умові / За бажання... and При бажанні?

Як обрати поміж:

За умови і При умові / За бажання... і При бажанні...?

Чи можна і так, і так?

3

Old school linguists (one, two) insist that only за is grammatical: за умови, за потреби, за бажання (or even бажаючи).
I would speculate that their reasoning is elimination of similarities with the Russian language, where при is used in this context.

However, classical literature has numerous examples of usage of при in the context of conditional relation.

Словник української мови, «при»:

  1. Is used to denote a phenomenon, event, etc. that acts as a condition for something
    Уживається з позначенням явища, події і т. ін., які виступають умовою чого-небудь

    При щирості між людьми, при глибокій і міцній симпатії ніякий риск не страшний! — Леся Українка
    Зітхне стара, при нагоді розповівши Тоні щось із тієї давньої давнини — Олесь Гончар

Note that the first example can be expanded like:

при щиростіпри умові щирості;

So, the answer would be, depending on your approach.
If you are a perfectionist, use only за умови.
If you allow yourself stepping off the classical rules, use both.

P.S. It seems to be impossible to provide with a short answer for the first part of the question, What's the difference between prepositions: “при” і “за”? as they both have pretty broad ranges of usage, and they overlap for only several contexts, including conditional relations.

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