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.