real analysis - Rigourous substitution formula for indefinite integral

By the fundemental theorem calculus we are able to say that:

The Substition Formula: If $f$ and $g'$ are continuous, then

$$\int_{g(a)}^{g(b)}f(u)du= \int_a^b f\big( g(x) \big)\cdot g'(x)dx$$

This formula is often stated for indefinite integrals by:

$$\int f(u)du= \int f\big( g(x) \big)\cdot g'(x)dx \quad \text{where} \quad u=g(x)$$

I was trying to find a rigorous proof to the second formulation, but have come up short so far. I was wondering whether it can indeed be rigorously proved, since it is often taught in early calculus class. I found a remark in Spivak's book on calculus saying that:

"This formula cannot be taken literally (after all, $\int f(u)du$ should mean a primitive of $f$ and the symbol $\int f\big( g(x) \big) g'(x)dx$ should mean a primitive of $(f\circ g)\cdot g'$; these are certainly not equal)."

Is there a rigorous sense in which the second formulation has a proof, or is just a symbolic manipulation?