The italian Wikipedia article about the fundamental theorem of calculus states it as such (I have summarized a bit):
First part: Let $f: [a,b] \to \mathbb{R}$ be a function continuous in the $(a,b)$ interval. Then its antiderivative $F$ exists in the interval and $F(x)=\int_a^x{f(t)dt}$. Moreover (corollary) if $f$ is continuous on $[a,b]$ and admits antiderivative $G$, then $\int_a^b{f(t)dt}=G(b) - G(a)$.
Second part: Let $f: [a,b] \to \mathbb{R}$ be a Riemann integrable function that admits an antiderivative $F$, then $\int_a^b{f(t)dt}=F(b) - F(a)$.
What confuses me a bit is the statement in the article that if $f$ is continuous then the first part can be derived from the second "using the basic properties of the derivative".
Can someone confirm this and show me how this works?
The existence of an antiderivative is needed as a hypothesis in the second part so how do we apply the second theorem? In the particular case in which $f$ is continuous I would rather derive the second from the first (if $f$ is continuous then it is also Riemann integrable).
I understand that the assumptions of the two parts are different, because in the first part we assume continuity and we do not assume the existence of an antiderivative, which is the thesis, while in the second part we do not assume continuity and we do assume the existence of the antiderivative, that in the thesis is used to calculate the definite integral.
Apologies to non italian readers, you will have to trust my translation:)