Let $E \in \mathcal{F}_Y$ and consider first the case where $f$ is an indicator function $I_E : Y \to \{ 0, 1 \}$. Applying results
we have
\begin{equation}
\int_Y I_E \, d \phi_* \mu
= \phi_* \mu(E)
= \mu(\phi^{-1} E)
= \int_X I_{\phi^{-1} E} \, d \mu
= \int_X (I_E \circ \phi) \, d \mu
\end{equation}
That is, the claim is true for measurable indicator functions. The full claim then follows by applying the principle of measure-theoretic induction. $\square$
