$\displaystyle Y_1, Y_2$ are independent IID samples from two normal populations sharing same variance

I can't show that the distribution of $\displaystyle (\frac{\overline{Y_1}-\overline{Y_2}}{S\sqrt{1/n_1+1/n_2}})=t_{(n_1+n_2-2)}$

What I did was to make use of the definition: $\displaystyle t_n\sim\frac{Z}{\sqrt{\chi ^2_n/n}}$

My Z is $\displaystyle \frac{(\overline{Y_1}-\overline{Y_2})-0}{\sigma\sqrt{1/n_1+1/n_2}}$

but I can't continue after this