in the section on soundness in the Metatheory chapter, there is what appears to be an orphaned reference to the principle of strong induction: "Note that this proof appeals to a principle of strong induction on the length of TFL-proofs. This is the first time we have seen that principle, and you should pause to confirm that it is, indeed, justified." Nothing else is said about induction in the textbook
forall x distinguishes between premises and assumptions, and so this proof should too