When trying to prove a logical equivalence, are you allowed to negate when you want?
For example am I allowed to do this
You can always substitute a statement with a logically equivalent statement if that's your question.
The two statements you have listed are not logically equivalent so you CANNOT substitute them for each other.
If you have an equivalence like then you are allowed to negate both sides and say because the statements are logically equivalent
(by logically equivalent I mean that if you wrote out truth tables, the values in the two columns would be identical)