Time-independent Schrodinger equation derivation

We can derive the Time-independent Schrodinger equation via Separation of variables as follows:

$$\begin{array}{rll}i\hslash \frac{\partial }{\partial t}|{\psi }_t⟩& =H|{\psi }_t)⟩& \text{plugging in the ansatz }|{\psi }_t⟩={\psi }_0(x)T(t)\\ i\hslash \frac{\partial }{\partial t}{\psi }_0(x)T(t)& =H{\psi }_0(x)T(t)& \\ i\hslash {\psi }_0(x)T^{\prime }(t)& =T(t)H{\psi }_0(x)& \\ i\hslash \frac{T^{\prime }(t)}{T(t)}& =\frac{H{\psi }_0(x)}{{\psi }_0(x)}& \end{array}$$

Since the left only depends on $t$ and the right on $x$, both sides must be constant. We call this constant the Energy $E$, and obtain the set of equations

$$\{\begin{array}{c}E=i\hslash \frac{T^{\prime }(t)}{T(t)}\\ E=\frac{H{\psi }_0(x)}{{\psi }_0(x)}\end{array}\Rightarrow \{\begin{array}{cc}T(t)\propto e^{-\frac{iEt}{\hslash }}& \\ E|\psi ⟩=H|\psi ⟩& \text{time-independent Schrodinger equation}\end{array}$$