Value of Term under Assignment Determined by Variables

Theorem
Let $\tau$ be a term of predicate logic.

Let $\mathcal A$ be a structure for predicate logic.

Let $\sigma, \sigma'$ be assignments for $\tau$ in $\mathcal A$ such that:


 * For each variable $x$ occurring in $\tau$, $\sigma \left({x}\right) = \sigma' \left({x}\right)$

Then:


 * $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma'}\right]$

where $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$ is the value of $\tau$ under $\sigma$.

Proof
Proceed by the Principle of Structural Induction applied to the definition of a term.

If $\tau = x$, then:

as desired.

If $\tau = f \left({\tau_1, \ldots, \tau_n}\right)$ and the induction hypothesis applies to each $\tau_i$, then:

The result follows from the Principle of Structural Induction.