Proof of change of variable in 2nd order PDE

Question

In a beginning text on PDEs we are given the equation on some $x$ interval $I$ with $t \geq 0$: $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}\dots(1)$$ The text then goes on to say that for any twice differentiable $F$, $$u(x,t)=F(x+t)$$ Is a solution to $(1)$. Now this seems intuitively fairly obvious to me but I want to be rigorous about it. To this end I say let $F(v)$ be twice differentiable. Thus $$\frac{d^2 F(v)}{d v^2}$$ exists (on what interval I'm not sure). We can now make the substitution $v=x+t$. Obviously $\frac{\partial x}{\partial v}=1=\frac{\partial t}{\partial v}$. By the chain rule then: $$\frac{d^2 F(v)}{dv^2}=\frac{\partial x}{\partial v}\frac{\partial }{\partial x}\frac{\partial x}{\partial v}\frac{\partial }{\partial x}F(x+t)=\frac{\partial^2 F(x+t)}{\partial x^2}$$ Similarly we have: $$\frac{d^2 F(v)}{dv^2}=\frac{\partial t}{\partial v}\frac{\partial }{\partial t}\frac{\partial t}{\partial v}\frac{\partial }{\partial t}F(x+t)=\frac{\partial^2 F(x+t)}{\partial t^2}$$ I'm not sure how rigorous this argument is though, and I'm not sure how to deal with the interval of $v$. Any criticism would be appreciated.