# Are read-once Boolean function in AC^0?

A Boolean function is said to be read-once if there is a Formula (a Boolean circuit where every gate has fanout at most $$1$$) computing the function $$f$$ such that every variable appears only once in the formula. Since each variable appears only once in the formula so it appears that this can be represented by a very small circuit of constant depth but I have a proof which contradicts my belief. I cannot get a feel of this statement.

(Proof of Read-once $$\notin AC^0$$)

Suppose read-once $$\in AC^0$$.

Consider a circuit in $$NC^1$$ and replace all the repeated literals with new variables $$y_1, y_2 \ldots y_t$$. Now this is a read-once function. We will now get the $$AC^0$$ circuit and replace back the $$y_i$$'s and get $$AC^0$$ circuit for an $$NC^1$$ circuit which shows that $$AC^0 = NC^1$$. Hence the contradiction.

I am still not getting a feel why read-once functions don't have small circuits. I will appreciate if you can give an example of read-once function for which I can't construct an $$AC^0$$ circuit.