Use the well-ordering principle to prove the induction on formulas. The induction on formulas theorem states:
(1) Every propositional variable has property $Q$.
(2) if $A$ is $\neg B$ and $B$ has property $Q$, then $A$ has property $Q$.
(3) If $A$ is one of $(B \vee C), (B \wedge C), (B \to C), (B \leftrightarrow C)$, and both $B, C$ have property $Q$, then $A$ has property $Q$.
Here is my approach: Assume (1)-(3) holds. Suppose there is some formula that does not have property $Q$. By well ordering principle, we can find such formula with least number of occurrences of the connectives. This means such formula is a single propositional variable that does not have property $Q$. However, according to principle (1), every propositional variable has property $Q.$ So we get a contradiction. So we are done.
However, I feel like this proof only uses the condition (1) and not concerning (2) and (3). Can anyone explain what is wrong in my approach?
I think that, in going from the assertion "we can find such formula with least number of occurrences of the connectives" to the next assertion "such formula is a single propositional variable", you overlooked the "such" in the first assertion. Note that it means "does not have property $Q$. Consider the possibility that all propositional variables have property $Q$ but some longer formulas don't. Then the formula with the fewest connectives and lacking property $Q$ would be some longer formula, not a single propositional variable. And that's where you'll need hypotheses (2) and (3) of the problem.