Step in proof of Ito formula

Question

I am reading a book on stochastic processes. The author proved Ito formula for $f(t,w(t))$ where $w(t)$ is brownian motion with filtration $F_t$. Then he wants to prove Ito formula for $x(t)=a(t)+b(t)w(t)$ where $a(t)=a(0)$ and $b(t)=b(0)$ and $a(0),b(o)$ are measurable with respect to $F_0$. The author considered a function $g(t,a_0,b_0,x)=f(t,a_0+b_0x)$, where $a_0,b_0$ are constants, then he writes that we can apply Ito formula for function $g$, and then he substitutes $a(t),b(t)$ instead of $a_0,b_0$. How can we justify this substitution? Can we use a formal definition of Ito integral, using integrals for step functions, or we can try conditioning on $F_0$?