Show that if A $\models_{\vec{x} / \vec{a}}$ $\forall x_i \phi$ then A $\models_{\vec{x} / \vec{a}}$ $\phi$.

Question

The setting for this problem is as follows:

Suppose that A $= \langle A,... \rangle$ is a structure for a language L and that the sequence $\vec{a}$ of elements of A interprets the variables $x_1,x_2,x_3,...$. Let $\phi$ be a formula of L in which $x_i$ does not appear as a free variable. Show that

if A $\models_{\vec{x} / \vec{a}}$ $\forall x_i \phi$ then A $\models_{\vec{x} / \vec{a}}$ $\phi$.

Answer 1

To show that if A $\models_{\vec{x} / \vec{a}}$ $\forall x_i \phi$ then A $\models_{\vec{x} / \vec{a}}$ $\phi$,

Recall that A $\models_{\vec{x} / \vec{a}}$ $\forall x_i \phi$ if and only if for all $b \in A$, A $\models_{\vec{x} / \vec{a} [x_i / b]}$ $\phi$.

Thus by taking $b = a_i$ (where $a_i$ is the element of the sequence $\vec{a}$ which corresponds to $x_i$)

we have that A $\models_{\vec{x} / \vec{a}}$ $\phi$.