Integration by Substitution/Primitive/Proof 1

Proof
Let $\map F x = \ds \int \map f x \rd x$.

Thus by definition $\map F x$ is a primitive of $\map f x$.

So $\map F x$ is an antiderivative of $\map f {\map \phi u} \dfrac \d {\d u} \map \phi u$.

Therefore:

where $x = \map \phi u$.