$[\![ \exists y \in \dot{x} \varphi(y) ]\!]$

$\displaystyle =\sup_{t}[\![ t \in \dot{x} \wedge \varphi(t) ]\!]$

$\displaystyle =\sup_{t}[\![ \varphi(t) ]\!] [\![ t \in \dot{x} ]\!]$

$\displaystyle =\sup_{t}[\![ \varphi(t) ]\!] \sup_{y \in \text{dom}(\dot{x})}[\![ y=t ]\!]\dot{x}(y)$

$\displaystyle =\sup_{y \in \text{dom}(\dot{x})}\dot{x}(y)\sup_{t}[\![ \varphi(t)\wedge y=t ]\!]$

$\displaystyle =\sup_{y \in \text{dom}(\dot{x})} \dot{x}(y)[\![ \varphi(y) ]\!]$