I am trying to prove that an equivalent way to perform the test for significance of regression in multiple linear regression is to base the test on $R^2$ as follows:
to test $H_0:\beta_1 = \beta_2 = ... = \beta_k$ versus $H_1$ : at least one $\beta_j \neq 0$, calculate $$F_0=\frac{R^2(n-p)}{k(1-R^2)}$$ and to reject $H_0$ if the computed value of $F_0$ exceeds $F_{\alpha,k,n-p}$, where $p = k + 1$.
Now, I am starting from the easier case of simple linear regression, and so I want to show that $$t^2=\frac{R^2(n-2)}{1-R^2}.$$ I did the following. $$t=\frac{\hat{\beta_1}}{se(\hat{\beta_1})}=\frac{S_{xy}}{S_{xx}}\sqrt{\frac{S_{xx}(n-2)}{SS_{res}}} \Rightarrow t^2= \frac{S^2_{xy}(n-2)}{S_{xx}SS_{res}}$$ and $$R^2=\frac{SS_{reg}}{SS_{tot}}=\frac{S^2_{xy}}{S_{xx}SS_{tot}} \Rightarrow 1-R^2=\frac{S_{xx}SS_{tot} - S^2_{xy}}{S_{xx}SS_{tot}} \Rightarrow \frac{R^2(n-2)}{1-R^2}=\frac{S^2_{xy}(n-2)}{S_{xx}SS_{tot} - S^2_{xy}}.$$
How do I conclude? Thanks!