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 $\tau$, $\map \sigma x = \map {\sigma'} x$

Then:


 * $\map {\operatorname{val}_\AA} \tau \sqbrk \sigma = \map {\operatorname{val}_\AA} \tau \sqbrk {\sigma'}$

where $\map {\operatorname{val}_\AA} \tau \sqbrk \sigma$ 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 $\tau = \map f {\tau_1, \ldots, \tau_n}$ and the induction hypothesis applies to each $\tau_i$, then:

The result follows from the Principle of Structural Induction.