How to show if $\vdash \alpha \leftrightarrow \beta$ then $\alpha \equiv \beta$

Question

We know that $\alpha \equiv \beta$ if $\beta \vDash \alpha$ and $\alpha \vDash \beta$.

In first order logic:

$(M, v) \vDash \alpha$ iff $(M, v) \vDash \beta$ for every interpretation $(M, v)$.

But how to show starting from $\vdash \alpha \leftrightarrow \beta$?

Answer 1

Suppose that $\alpha\leftrightarrow\beta$ is a valid sentence. Assume that $(M,v)\models\alpha$, since $\alpha\leftrightarrow\beta$ is a valid sentence, it must be the case that $(M,v)\models\alpha\leftrightarrow\beta$...

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