Taking the definition of vacuous truth to be an implication where nothing satisfies the antecedent. Is there notation commonly used for "non-vacuous implication"?
I could write:
$(\forall x . P(x) \implies Q(x)) \land (\exists x . P(x))$
But I need to write it a lot, so I would prefer to write some shorthand.
You may want to consider using a $\LaTeX$ command such as \xrightarrow to create a symbol such as $\xrightarrow[]{\exists}$ or $\xrightarrow[]{\text{NV}}$ to denote 'non-vacuous implications'. For example, consider writing $$P(x) \xrightarrow[]{\text{NV}} Q(x)$$ in place of $\left(\forall x \ P(x) \rightarrow Q(x)\right) \wedge \left(\exists x \ P(x) \right)$.
Several Google searches concerning non-vacuous implications suggest to me that there may not be any notation commonly used for non-vacuous implications.
