Consider a sequence $\alpha_1,\alpha_2,...$ of wffs. For each wff $\varphi$ let $\varphi^*$ be the result of replacing the sentence symbol $A_n$ by $\alpha_n$, for each $n$.
(a) Let $v$ be a truth assignment for the set of all sentence symbols; define $u$ to be the truth assignment for which $u(A_n)=\bar{v}(\alpha_n)$. Show that $\bar{u}(\varphi)=\bar{v}(\varphi^*)$. Use the induction principle.
(b) Show that if $\varphi$ is a tautology, then so is $\varphi^*$.
Now, I got the part (a), but I can't induce part (b) from part (a), because of truth assignments for $\varphi^*$. Part (a) doesn't induce all truth assignments for $\varphi^*$, but only that for $\varphi$ and the corresponding $\bar{v}(\alpha_n)$. Is it enough to just care about $\bar{v}(\alpha_n)$'s to show $\varphi^*$ is a tautology?
Let $v$ an assignment whatever.
We have to consider the corresponding assignment $u$ such that $u (A_n)=\overline v(\alpha_n)$.
By the previous result: $\overline u(\varphi)=\overline v (\varphi^*)$.
But $\varphi$ is a tautology; thus, $\overline u(\varphi)= \text T$.