Suppose a bar of length $L$, Young Modulus $Y$, temperature coefficient $\alpha$ is at temperature $T_o$ is just held between two walls. Now temperature increases, to make a change in temperature $\triangle T$ (measured from initial $T_o$). My book says thermal stress developed is $Y\alpha\triangle T$.

My question:

As the rod is rigidly held, walls will not let rod to changes its length further from $L$, for this it applies force $F$. Its original length at $T_o+\triangle T$ would have been $L(1+\alpha\triangle T)$, if walls were not present, hence it causes $\triangle L=l\alpha\triangle T$.

So $Y=\dfrac{F_{T_f}\cdot L_{T_f}}{A\cdot\triangle L}\tag*{1}$ where $T_f$ signifies their respective values at final temperature. Putting these it gives to me themal stress as, $$\sigma=\dfrac{Y\alpha\triangle T}{1+\alpha\triangle T} $$.

If I were to put $L_{T_f}=L$ then this gives what my book says, but we should apply $\text{eq. 1}$ and put everything at the same temperature, so why putting its length as $L$ mandatory. Please help.