Prove Law of Chemical Equivalence
An example

Here is a balanced equation:

$$\ce{2 H2 + O2 -&gt; 2 H2O}$$

Both sides of the equation show four atoms of hydrogen and two atoms of oxygen, so it is balanced. The coefficients are $\nu_{\ce{H2}} = 2$, $\nu_{\ce{O2}} = 1$ and $\nu_{\ce{H2O}} = 2$.

If 20 moles of $\ce{H2}$ and 10 moles of $\ce{O2}$ are used up in a reaction forming water (you can figure out 20 moles are formed), I can write:

$$\frac{n_{\ce{H2}}}{\nu_{\ce{H2}}} = \frac{n_{\ce{O2}}}{\nu_{\ce{O2}}},$$ where $n$ is the amount that is used up in the reaction.

I can not write:

$$n_{\ce{H2}} \cdot \nu_{\ce{H2}} = n_{\ce{H2}} \cdot \nu_{\ce{H2}},$$

as you will find when substituting the values from the example.

I should also not write:

$$2 \cdot \ce{H2} = 1 \cdot \ce{O2} $$

Hydrogen atoms are not oxygen atoms, no matter what we multiply it (nuclear reactions aside)

What about $I$ and $i$?

If you call the stoichiometric coefficients $i$ instead of $\nu$, and you want the equation below to be true:

$$i_{\ce{H2}} \cdot I_{\ce{H2}} =  i_{\ce{O2}} \cdot I_{\ce{O2}},$$

you have to make sure that $I$ is proportional to $\frac{1}{i}$, i.e. define them by 

$$I_{\ce{H2}} = \frac{ \mathrm{const} }{ i_{\ce{H2}} } $$

$$I_{\ce{O2}} = \frac{ \mathrm{const} }{ i_{\ce{O2}} }$$

If you look at definitions of "number of equivalents" (a concept that is outdated), you will find that this is the case. Once that is in place, it is a matter of showing that for any non-zero $a$ or $b$, 

$$\mathrm{const} \cdot \frac{a}{a} = \mathrm{const} \cdot \frac{b}{b} $$


  


Source: https://www.askiitians.com/iit-jee-chemistry/physical-chemistry/stoichiometry-and-redox-reactions/law-of-equivalents.aspx