Sum of RVs equal in distribution implies RVs are If $X+Y \stackrel{d}{=} X^{'}+Y$, with $Y \sim N(0,1)$ independent of both $X$ and $X^{'}$, is there any example in which $X$ and $X^{'}$ are not equal in distribution?...--prophetes.ai

Sum of RVs equal in distribution implies RVs are If $X+Y \stackrel{d}{=} X^{'}+Y$, with $Y \sim N(0,1)$ independent of both $X$ and $X^{'}$, is there any example in which $X$ and $X^{'}$ are not equal in distribution? It seems to me that: $$\phi_{X+Y}(t)=Ee^{itX}Ee^{itY}$$ $$\phi_{X^{'}+Y}(t)=Ee^{itX^{'}}Ee^{itY}$$ Therefore, $Ee^{itX}=Ee^{itX^{'}}$, and so $X \stackrel{d}{=} X^{'}$

Your argument is basically right, but it leaves out the well-known fact that $\phi_Y(t)=Ee^{itY}=\exp(-t^2/2)$ never vanishes so the step from $\phi_{X+Y}=\phi_{X'+Y}$ to $\phi_X = \phi_{X'}$ is valid.