Lemma - Protasis :
Consider the PDE $u_t + a(u)u_x = 0$. Now, let $A'(u) = a(u)$. The PDE expression can be written as $u_t + A(u)_x = 0$. For this equation to hold, under the sense of distributions, it means that : $$\int_0^\infty\int_{-\infty}^\infty[u\psi_t +A(u)\psi_x]\mathrm{d}x\mathrm{d}t=0$$ This expression is equivalent to : $$\frac{\mathrm{d}}{\mathrm{d}t}\int_a^bu(x,t)\mathrm{d}x+A(u(b,t))-A(u(a,t))=0,\quad \forall a,b$$
I would like to request a proof to the protasis above if possible, which means a proof for the equivalence of the $2$ expressions.
(This is mentioned as a theoritical exercise-protasis proof in "Walter A. Strauss - An Introduction to PDEs")