Assume, $F_0:[a,b]\to\mathbb R$ is an antiderivative of $f$, i.e. by definition $$F_0(x)=\int_a^xf(t)dt.$$
It follows from the definition of the Riemann integral that $$F_0(a)=0,\quad F_0(b)=\int_a^bf(t)dt.$$
For an arbitrary antiderivative $F$ of $f$ it follows from the fact that antiderivatives are unique up to a constant that $$F(b)-F(a)=F_0(b)-F_0(a)=F_0(b)=\int_a^bf(t)dt.$$
