Fundamental lemma of variational calculus for Bochner space

Question

So I was wondering if there is a "nice" way to prove a variant of the fundamental lemma of variational calc. The setting is as follows: Let $n \in \mathbb{N}$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain with $C^1-$boundary and $T>0$. Let $$f \in L^{\infty}((0,T);L^2 (\Omega)) \cap L^2 ((0,T); \overset{\circ}{{H}^1}(\Omega))$$ where $L^p(X;Y)$ denotes the space of Bochner $p$-integrable functions (so $Y$ is a Banach space) and $H^1(X;Y)$ the Sobolev space $W^{1,2} (X;Y)$. Then, if for all $\varphi \in C_0^{\infty}((0,T) \times \Omega)$ (functions with compact support) with $\Omega_T := (0,T) \times \Omega$: $$ \int_{\Omega_T} f\cdot \varphi d(x,t) =0$$ it follows that $f=0$ a.e.