For convenience, let us define a wff to be positive if there is no use of the negation symbol $\neg$ at all in the wff. Hence, for example, $W=P\iff Q$ is a positive wff.
Now the question is to show that if $W$ is any wff built from propositional variables $P_1,\dots,P_n$ such that if $V(P_1)=\cdots=V(P_n)=T$ implies $V(W)=T$ then $W$ is logically equivalent to a positive wff.
I tried this with an example:
Let $W=\neg[(P\implies \neg Q)\land(R\land(P\lor Q))]$. Then when $V(P)=V(Q)=V(R)=T$ we also have that $V(W)=T$, and in fact we can show (by directly using simple logical equivalences) that $W$ is logically equivalent to $R\implies (P\iff Q)$, which is a positive wff by the definition above.
But for the general case, I haven't got a clue. I tried induction on the length of $W$ to no success. Also tried induction on number of propositional variables, again with no success.
Can anyone provide hints?