Exercise $5\mathrm{.}1\mathrm{.}3.$ Mimic the proof of Generalization Theorem to show the fol$-$

lowing Generalization on Constant Theorem, which will be used in the proof of

Completeness Theorem. Let $,c$ and $y$ be a formula, a constant and a variable

respectively. We $($ab$)$use $[c$:$=y]$ to denote the formula obtained from $$ by

substituting $y$ for all occurrences of $c.$

$($a$)$ Show that if $$ is an axiom, then $[c$:$=y]$ is also an axiom. Hint: Check

that for any term $t,([x$:$=t])[c$:$=y]$ is $([c$:$=y])[x$:$=t[c$:$=y]],$ where

$t[c$:$=y]$ is the term obtained from $t$ by substituting $y$ for all occurrences

of $c.$

$($b$)$ Generalization on Constants Theorem Assume that $|--$ and that

$c$ is a constant symbol which does not occur in $.$ Then there is a variable

$y,$ which does not occur in $,$ such that $|--[c$:$=y].$ Furthermore, there

is a deduction of AAy$[c$:$=y]$ from $$ in which $c$ does not occur.