Exercise 7.6 [deduction-theorem-exercise]

Prove each of the following assertions:

1. $\alpha$ is valid if and only if ${True}{\models}\alpha$.

2. For any $\alpha$, ${False}{\models}\alpha$.

3. $\alpha{\models}\beta$ if and only if the sentence $(\alpha {:\;{\Rightarrow}:\;}\beta)$ is valid.

4. $\alpha \equiv \beta$ if and only if the sentence $(\alpha{\;\;{\Leftrightarrow}\;\;}\beta)$ is valid.

5. $\alpha{\models}\beta$ if and only if the sentence $(\alpha \land \lnot \beta)$ is unsatisfiable.