It seems that only quantifier-free formulas are considered here.

You cannot assume that and are atomic formulas (does it mean the same as prime formulas?) in the induction step. E.g., where is a prime formula has the form , but is not prime. I assume you can finish the definition and the proof that iff .

I believe this holds for any truth assignment v. Call this fact (*).

Suppose for every in . Let be the truth assignment constructed from and as described in (a). Then by (a), so by putting in (*) we have , which again by (a) implies .