Does someone know where this theorem is from? From which statistic book?
I've searched on two books already: Wackerly and Canavos statistic books.
Theorem. Let $x_i$ and $y_i$ two independent samples of size $i=1,n$ and $j=1,m$; $n,m\ge30$ both d.i.i.s.r.s and for $(1-\alpha)100 \%$ of confidence interval we have that $$\mu_x-\mu_y\in[(\overline x-\overline y)-z_\frac{\alpha}{2}\sqrt{S^2_x/n+S^2_y/m},(\overline x-\overline y)+z_\frac{\alpha}{2}\sqrt{S^2_x/n+S^2_y/m}]$$
Result of a CLT, Slutsky and some technical requirements.
$\sqrt{n}(\bar{X}_n - \mu_x) \to N(0, \sigma_x^2)$ as $n\to \infty$, and $\sqrt{m}(\bar{Y}_m - \mu_x) \to N(0, \sigma_y^2)$ as $m \to \infty$. Now, for finite $m$ and $n$ we have $\bar{X}_n - \mu_X \sim^{approx.}N(0, \sigma_X^2/n),$ $\bar{Y}_m - \mu_Y \sim^{approx.}N(0, \sigma_Y^2/m)$, $$ (\bar{X}_n - \mu_X) + (\mu_Y - \bar{Y}_m) \sim^{approx.} \sqrt{ \sigma_X^2/n + \sigma_Y^2/m}N(0,1) , $$ hence, $$ \frac{(\bar{Y}_m - \bar{X}_n ) - (\mu_Y - \mu_X) }{\sqrt{S_X^2/n + S_Y^2/m}} \sim^{approx.} N(0,1) . $$