looking through Nicolas Privault's Introduction to Stochastic Finance with Market Examples, I ran into the following version of Levy-Khintchine formula: $$ \mathbb E\left[\exp\left(\int_0^T f(t) dY_t\right) \right] = \exp\left(\lambda \int_0^T \int_{-\infty}^{\infty}(e^{yf(t)}-1)\nu(dy)dt\right). \tag 1 $$
Here $Y_t$ is a compound Poisson process defined as $$ Y_t = \sum_{k=1}^{N_t}Z_k $$ where $N_t$ is a simple Poisson process with intensity $\lambda$. Random variables $Z_k$ are independent, identically distributed, square-integrable and their probability distribution is $\nu(dy)$.
The book claims that formula $(1)$ follows as a consequence of the following proposition: $$ \mathbb E[e^{\alpha(Y_T -Y_t)}] = \exp ((T-t)\lambda (\mathbb E[e^{\alpha Z_1}]-1)). $$
It is not immediately clear how formula $(1)$ immediately follows from the above proposition. I would appreciate any help.