Proof of validity of tautology in first order logic Every first-order logic formula which has a tautological shape in propositional logic is a valid formula. Is it possible to give  a formal proof for the above?

The only way I can make sense of "tautological shape" is to suppose a formula $\psi$ is of the form: $$\psi = \alpha[\phi_1 / p_1, \dots,\phi_n/p_n]$$
where $/$ denotes substitution (adequately defined), for some first-order formulas $\phi_i$ and a propositional tautology $\alpha$ in the variables $p_i$.
Now let $\mathcal M$ be a structure for $\psi$ and let $v: {\sf Var} \to \mathcal M$ be an assignment of variables, both arbitrary.
The crux is to define the boolean interpretation $v': \{p_1,\dots,p_n\} \to \{T,F\}$ by: $$v'(p_i) = \begin{cases}T&\text{ if $\mathcal M,v \models \phi_i$}\\F&\text{ if $\mathcal M, v \not\models \phi_i$}\end{cases}$$
and subsequently demonstrate by induction on the complexity of propositional formulas $\beta$ in the symbols $p_i$ that: $$v'(\beta) = T \iff \mathcal M,v\models \beta[\phi_i/p_i]$$
Having established this equivalence, since $\alpha$ is a tautology, we conclude that $\mathcal M,v \models \psi$ for all structures $\mathcal M$ and assignments $v$; that is, $\psi$ is valid.
