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What exactly are the identity rules in logic?

In first order logic, I have read there are a couple of identity rules and I have a question for example: If you have "a=b" does it mean that I can also write it as "b=a"? Is it true the one-way or both? And if I have two statements such as "a=b" and "a=c". Can I derive both "b=c" or "c=b" from this?