It's true actually. As a hint, the Euclidean property $$xRy \quad\&\quad xRz \implies yRz$$ does not require the $x,y,z$ to be distinct. So for instance it implies $$xRy \quad\&\quad xRy \implies yRy.$$

It's true actually. As a hint, the Euclidean property $$xRy \quad\&\quad xRz \implies yRz$$ does not require the $x,y,z$ to be distinct. So for instance it implies $$xRy \quad\&\quad xRy \implies yRy.$$

Try to construct a counterexample model with 4 worlds $w,x,y,z$ where $\phi$ holds only in $z$ and where $wRx, xRy, yRz$.