All textbooks I've come across have the same way of proving the mirror formula. Everything makes sense, except the fact that they apply the sign convention once to obtain the mirror/lens formula, but when they use the formula to solve a question, they apply the sign convention again. Why?
This double application of the convention adopted is fundamental to use the formula you derived to make useful predictions.
In fact, quantities involved in such demonstrations (I assume you are talking about geometrical optic, aren't you?) need a convention in order to account for the reciprocal position of two points. If you stated that some quantity is positive in a certain direction in the demonstration of a certain formula, then when it comes to apply this formula you necessarily have to use the same convention, in order to this formula to be valid and useful to make predictions (i.e., solve exercises).