Help me proving Dalton's law of partial pressure formula How do I prove 


  $${{\displaystyle p_{i}=p_{\text{total}}y_{i}}}$$ 


where $y_i$ is the mole fraction of the $i^{th}$ component in the total mixture of $n$ components ?

Here is what I know

Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation:

${\displaystyle p_{\text{total}}=\sum _{i=1}^{n}p_{i}}$

or       

${\displaystyle p_{\text{total}}=p_{1}+p_{2}+p_{3}+...p_{n}}$

And mole fraction $y_i= \dfrac{n_i}{n_{\text {total}}}$

I don't see how to relate , please help.
After having some discussion with @JohnRennie 

Let $P_i=n_i \dfrac{RT}{V}$ then $P_i=\dfrac{n_i}{n_{\text{total}}}\dfrac{n_{\text{total}}RT}V$

$P_i=x_i P_{\text{total}}$

Where $x_i$ is mole fraction of $i^{\text{th}}$ particle.