Check list for strings using string?

Hi I am having a little trouble testing a list to see if the list contains only strings. I am reading the documentation 2.3(Iterations, and Recursion) + 3.12 (Conditionals) and I can't quite seem to work lists in the way that I want.

I'm trying to keep it as simple as possible,

(define (strings-only? lst)
  [cond
    [(empty? lst) true]
    [else (string?(first lst))
          (string?(rest lst))]])

What I assume this does is first checks the list (lst) to see if it's empty. Then it checks the first variable of the list to see if its a string, then checks the rest. When I put in

(define a (list "This" "that"))

It pops back that the test is false. I feel like the problem is with the way I'm trying to iterate through the list with first/rest. Any thoughts would be appreciated.