I have private health insurance and insurance from work?

I have private health insurance from blue cross blue shield, I also get blue cross blue-shield from the work I do for the state. If I was to use the insurance from my work would it be a conflict with the insurance I privately have? I was told that they would cancel each other out and I would lose my private insurance. Is this true??? I don’t really want to lose my private insurance for health and security reasons but I also don’t want to keep paying more if i don’t have to in the long run.

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