[x]_1=[y]_2

\Longrightarrow (x \rho _1 y) i (x \rho _2 y)

\Longrightarrow [x]_1=[y]_2=[x]_2

\Longrightarrow (\forall x)(\forall a) (x \rho _1 a \iff x \rho _2 a)

\Longrightarrow \rho _1=\rho _2