Same inner product with functions in $C^{\infty}_c(\Omega)$ implies same functions in $H^1(\Omega)$?

Let $\Omega\subset\mathbb{R}^n$ be an open set. It is true that if $u,v\in H^1(\Omega)$ and$$(u,\phi)_{H^1(\Omega)}=(v,\phi)_{H^1(\Omega)},\ \forall\phi\in C^{\infty}_c(\Omega)$$then $u=v$? Here I have denoted by $(\cdot,\cdot)$ the inner product and $C^{\infty}_c(\Omega)$ the smooth functions with compact support in $\Omega$. I know that this statement its true when the inner product its in $L^2(\Omega)$ but I dont know if its true when the inner product its is in $H^1(\Omega)$.

Thanks for the help !