Let $\Omega \subset \mathbb{R}^n$ be an open bounded set, $Q_T:=\Omega \times (0,T)$ and $f \in L^2(Q_T)$. Suppose that
$$(a)\;\; \int_{Q_T} \partial_t u \;\phi+\int_{Q_T} \nabla u \; . \nabla \phi = \int_{Q_T} f \phi \;\; \forall \phi \in L^2(0,T;W^{1,\infty}(\Omega))$$ where $\partial_t u \in L^2(0,T;{W^{1,\infty}}(\Omega)^*)$
Prove that $ \forall \phi \in L^2(0,T;H^1(\Omega) \cap L^\infty(\Omega))$
$$ (b) \;\;\;\int_{Q_T} \partial_t u \;\phi+\int_{Q_T} \nabla u \; . \nabla \phi = \int_{Q_T} f \phi. \;\;$$
My attempt :
Let $\phi \in L^2(0,T;H^1(\Omega) \cap L^\infty(\Omega))$, and consider $$ A(\phi):=\begin{cases} \phi \;\;\;\;\; \text{if}\;\;\; \vert \phi(x,t)\vert \leq \Vert \phi\Vert_{L^\infty(\Omega)}\\ \Vert \phi \Vert_{L^\infty(\Omega)} \;\;\;\;\; \text{if}\;\; \Vert \phi\Vert_{L^\infty(\Omega)} \leq \phi(x,t)\\ -\Vert \phi \Vert_{L^\infty(\Omega)} \;\;\;\;\; \text{if}\;\; \phi(x,t) \leq -\Vert \phi\Vert_{L^\infty(\Omega)}\\ \end{cases} $$ then choosing $A(\phi)$ as a test function in $(a)$, we obtain $(b)$.
Is this approach correct?