= Solution
<Semantic consequence under an assignment> means that for every <first-order structure> $\mathcal A$ with the relevant <first-order variable assignment> $v$ into its domain,
$$
\boxed{\bigl(\mathcal A\models\gamma[v]\text{ for every }\gamma\in\Gamma\bigr)\ \Longrightarrow\ \mathcal A\models\phi[v].}
$$
More explicitly, in a fixed structure this is the satisfaction implication at $v$; logical consequence requires that implication in all structures and assignments. An assignment takes values in a particular domain, so a single set map is not silently shared between unrelated domains.
By induction on <first-order formulas>, their <truth values> depend only on the values of their <free variables>. A <first-order sentence> has none. If all formulas here are sentences, every displayed satisfaction statement, and therefore the consequence condition, is independent of $v$.
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