Proof of Strong Markov Property in Liggett

I encountered a problem while having a look at the proof of the strong Markov Property in Liggett's book about Markov Processes "Continuous Time Markov Processes: An Introduction". I want to proof the Strong Markov Property for continuous Markov chains using the same approach as in the book. As far as I can see the proof in the book uses the following statement:

Let $(X_t)_{t\geq 0}$ be a continuous Markov chain with values in the discrete space $S$. Further let $Y$ be a bounded random variable and $\xi$ a finite stopping time. Then the following holds $Pr_x$-a.s.

$$E_x[Y] = E_{x}[E_{\xi}[Y]] \iff E_x[E_x[Y\cdot \theta_{\xi} \vert {\cal{F}}_{\xi} ]] = E_{x}[E_{\xi}[Y]]$$

$$\implies E_x[Y\cdot \theta_{\xi} \vert {\cal{F}}_{\xi} ] = E_{\xi}[Y]$$

where ${\cal{F}}_{\xi} := \{ A \in {\cal{F}}: A \cap \{\xi\leq t\}\in {\cal{F}}_t \text{ for all } t\geq 0 \}$ and $\theta_s$ the shift operator for right-continuous functions $\omega$ i.e. $(\theta_s \omega)(t) = \omega (t+s)$.

I have a problem with the implication in the second line of the equations. In general this doesn't hold because I can find two different random variables s.t. the expectation would be the same, right? So am I missing another argument that is being used to proof the statement?

Thank you for your help!