I am trying to prove myself that stepwise method should not be used. Indeed we are often modeling data likewise at my work. I have recently bought the very interesting book of Frank Harrell (Regression Modeling Strategies). In section 4.3 Variable selection he states the following:
But using $Y$ to compute $P$-values to decide which variables to include is similar to using $Y$ to pool treatment in a five-treatment randomized trial, and then testing for global treatment differences using fewer than four degrees of freedom.
He gave a similar explanation in a post here on CrossValidated but I do not get both (pooling then testing for global differences).
I understand that there is a problem of tests multiplicity but I would like to have a more technical proof or more details regarding these examples.