Doubt with proof of integral of odd function

[Hernaner28]

1. The proof

2. The doubt

What I don't understand is when he switch the variable back to x. He says that:
$$\displaystyle \int_{-a}^{0}{f(x)dx=-\int_{0}^{a}{f(x)dx}}$$

But if we have:
$$\displaystyle \int_{0}^{a}{f(-t)dt=-\int_{0}^{a}{f(t)dt}}$$

Then if we switch back to x we should have:
$$\displaystyle \int_{-a}^{0}{f(x)dx=\int_{-a}^{0}{f(-x)dx}}$$

And not what he said. Could you clarify that to me?

Thanks!

 

[Hernaner28]

I think I got it. I think is a matter of the name of the variable.