This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. There must be some trick that I am missing.

A forcing notion $P$ is $RUCar^{V}$-semiproper iff there is a cardinal $\kappa$, such that for all cardinal $\lambda \geq \kappa$, for all $N$ countable elementary submodel of $(H(\lambda), \in)$ with $P \in N$, if $p \in P \cap N$ then there is $q$ ($\in P$) extending $p$ satisfying the following:

for every regular uncountable cardinal $\theta \in N$ and $P$-name $\pi \in N$ of an element of $\theta$, $q \Vdash_{P}$ "there is an ordinal $\alpha \in N$ with $\pi < \alpha < \theta$".

Shelah claims every $RUCar^{V}$-semiproper forcing notion is proper. Is it really easily observable?