Question about Succession Wars

Hello, me again with another (probably super-noob question).

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Playing as PLC (c. 1618), just waiting around for the Religious League war to start, when I get the notification that my two most hated enemies - Bohemiablob (which had eaten Hungary) and a (super-weakened) Muscovy were about to become a personal union! Time for my first ever succession war.

I was given the option to declare war to intervene, and I said "aw hell nah, I'm not letting Bohemia get away with this!" and declared. I had over 100k troops with aristocratic, quantity, and defensive ideas, and I was equal in tech to Bohemia and two ahead of Muscovy. I figured my army would steamroll Bohemia's 40k and Muscovy's pathetic 17k, right? Bohemia's only allies were shitty HRE countries (and Ulm).

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Next thing I know, I'm at war with the Ottomans, Sweden, and France, facing down over 220k troops. My allies Austria and Brandenburg wouldn't join ("they are the Holy Roman Emperor" and "they are allied to the Emperor"). The only way I could force a stalemate was with the help of superpower Spain and the Mamluks, but now the League War will fire at any minute and I'm down 110k manpower, so it's not ideal.

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So... how do succession wars work? Why does the emperor refuse to get involved while literally everyone else jumps in against me?

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Thanks in advance for the help!