A computation for a continuous-time Markov chain

Let $P^{y}$ be a continuous-time Markov chain with transition matrix G, started at y (it is in a finite state space, it is irreducible, positive recurrent). And let T be an exponential random variable of parameter 1 independent of $P^{y}$. We also define $T_{\lambda}=T/\lambda$ for all $\lambda>0$. We define $\phi^{\lambda}(y)=\mathbb{E}\left[\int_{0}^{T_{\lambda}}f(P_{t}^{y})dt\right]$. How to prove that $(G-\lambda)\phi^{\lambda}=-f$ ?

I guess we should condition on $T_{\lambda}$ and use the independence of T, but I can't find a way to do it properly.

Thanks in advance for any suggestion :)

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