This is another exercise problem (Exercise 3 of Section 2.4) of Enderton's "A mathematical introduction to logic".

(a) Let $\displaystyle A$ be a structure and let $\displaystyle s:V \rightarrow |A|$. Define a truth assignment $\displaystyle v$ on the set of prime formulas by $\displaystyle v(\alpha)=T$ iff $\displaystyle \models_A \alpha[s]$. Show that for any formula (prime or not), $\displaystyle \bar{v}(\alpha) = T$ iff $\displaystyle \models_A \alpha[s]$.

(b) Conclude that if $\displaystyle \Gamma$ tautologically implies $\displaystyle \phi$, then $\displaystyle \Gamma$ logically implies $\displaystyle \phi$.

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What I have tried so far is

For (a), if $\displaystyle \alpha$ is a prime formula, nothing needs to be done. If $\displaystyle \alpha$ is not a prime formula, two cases have to be considered, i.e., $\displaystyle \alpha = \neg \beta$ or $\displaystyle \alpha = (\beta \rightarrow \gamma)$, where $\displaystyle \beta$ and $\displaystyle \gamma$ are both atomic formulas.

For (b), if $\displaystyle \bar{v}(\psi) = T$ for every $\displaystyle \psi \in \Gamma$, then $\displaystyle \bar{v}(\phi)=T$ by assumption. Then I need to show if $\displaystyle \models_A \psi[s]$, then $\displaystyle \models_A \phi[s]$. I am stuck here.

Any help will be appreciated.