Coincidence Lemma (First-Order Logic)

first-order-logic

Definition

Coincidence Lemma (First-Order Logic)

Let $\phi$ be a formula and let $\ell ,{\ell }^{\prime }$ be two variable assignments that agree on $\mathrm{free}(\phi )$. Then

$$(\mathcal{M},\ell )\models \phi ⟺(\mathcal{M},{\ell }^{\prime })\models \phi$$

Satisfaction depends only on the free variables. Bound variables do not affect the truth value.