Let $A$, $B$ be two densely defined operators on a Hilbert space $H$. Suppose the domains obey $D(B) \supseteq D(A)$, and suppose $A$ is a closed operator with $\mathcal{D} \subseteq D(A)$ a core for $A$ (i.e., $\overline{A\rvert_{\mathcal{D}}} = A$). Assume that for some $C > 0$,
$$\|B \varphi \| \le C\|A\varphi \| + C\|\varphi\|, \qquad \text{all $\varphi \in \mathcal{D}$} \tag{1} \label{est on core}.$$
I would like to show that the estimate \eqref{est on core} holds more generally for any $\varphi \in D(A)$ (not just for $\varphi \in \mathcal{D}$).
I can prove this result under the additional assumption that $B$ is a symmetric operator. However, according to page 162 of Reed and Simon vol. 2 (which leads into the proof of the Kato-Rellich Theorem), the result holds without this assumption.
For arbitrary $\varphi \in D(A)$, using that $\overline{A\rvert_{\mathcal{D}}} = A$, we get a sequence $\psi_j \in \mathcal{D}$ so that
$$\psi_j \to \varphi, \, A \psi_j \to A \varphi.$$
We have
$$\|B\varphi\| = \sup_{\eta \in D(A) : \|\eta\|= 1} |\langle B \varphi, \eta \rangle| = \sup_{\eta \in D(A) : \|\eta\|= 1} |\langle \varphi, B\eta \rangle|, $$
by symmetricity of $B$. But for each $\eta \in D(A)$ with $\|\eta\| = 1$, using symmetricity again.
$$ |\langle \varphi, B\eta \rangle| = \lim_{j \to \infty} |\langle \psi_j, B\eta \rangle| = \lim_{j \to \infty} |\langle B \psi_j, \eta \rangle| \le \lim_{j \to \infty} C\|A\psi_j| + C\|\psi_j\| = C\|A\varphi| + C\|\varphi\|. $$
Is there a way to reach the same conclusion without assuming $B$ is symmetric? Hints or solutions are greatly appreciated.
The symmetricity of $B$ is not needed. Since $A$ is assumed to be self-adjoint, it is a closed operator, hence complete with respect the to "graph" norm $$ \|\varphi_1\|_1 = a \| A \varphi \| + b\| \varphi \|.$$
So, by the BLT theorem, the estimate will hold for all of $D(A)$ if it holds on a core for $A$.