How to prove the following theorem using induction on formulas?
Let V and V' be two valuations of L. Let $\alpha$ be a formula such that V(p) = V'(p), for all atomic formula p that is subformula of $\alpha$. Then V($\alpha$) = V'($\alpha$).
The base case is simple: If $\alpha$ is an atomic formula, then it does not have a subformula so V($\alpha$) = V'($\alpha$). But how to proceed using induction on formulas?