Do you want to know how to write "let" exactly or basically write "the same" content as how we usually interpret "let"?
I don't know your particular proof, but you might achieve the same effect by considering conditionals. In other words, "let x equal a" turns into a conditional "if x equals a, then ...", at least in a system which has the material conditional at work. You could write ((x=a)->...) or with (x=a) as a proposition p, (p->...). If you don't have the material condtiional, I'd guess you'd want to use one of its respective equivalents... such as in a deductive system with only disjunction and negation, instead of saying "if a, then b" you would have "either not a or b".