# OLS - Predeterminedness and moment condition

I'm having trouble validating if following procedure to test for predeterminedness is plausible. Given the linear model:

$$y_t=\beta_1+\beta_2x_{1t}+\beta_3x_{2t}+\epsilon_t$$

Having $$x_{2t} = y_{t+1|t}+v_t$$ and $$\epsilon_t$$ uncorrelated with $$x_{1t}, x_{2t}$$. And $$\epsilon_t$$ i.i.d. with $$E(\epsilon_t)=0, E(\epsilon_t^2)=0$$. Given $$v_t$$ i.i.d. and also uncorrelated with $$\epsilon_t$$.

$$$$\label{eq1} \begin{split} E[x_{1t}\epsilon_t] & = E[E[x_{1t}\epsilon_t|x_{1t}]] \\ & = E[x_{1t}\underbrace{E[\epsilon_t|x_{1t}]]}_{=0} = 0 \\ E[x_{2t}\epsilon_t] & = E[E[x_{2t}\epsilon_t|x_{2t}]] \\ & = E[x_{1t}\underbrace{E[\epsilon_t|x_{2t}]]}_{=0} = 0 \\ \end{split}$$$$

Can anyone confirm this?