 Semantic consequence under an assignment

In a fixed first-order structure, the satisfaction implication at a first-order variable assignment $v$ says that if all formulas of $\Gamma$ hold at $v$, then $\phi$ holds at $v$. Semantic consequence requires that implication in every relevant structure and assignment. The assignment's codomain belongs to its structure; different domains are not silently assigned one common map. When all formulas are first-order sentences, assignments do not affect any of their truth values.