Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations

Theorem
Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.

The following definitions of semantic equivalence for boolean interpretations of $\mathbf A$ and $\mathbf B$ are equivalent:

Definition 1 implies Definition 2
Let $\mathbf A, \mathbf B$ be equivalent according to definition 1.

Let $v$ be a boolean interpretation.

By definition. either $v \left({\mathbf A}\right) = T$ or $v \left({\mathbf A}\right) = F$.

In the first case, it follows by hypothesis that $v \left({\mathbf B}\right) = T$.

In particular, then:


 * $v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

In the second case, it must be that $v \left({\mathbf B}\right) \ne T$.

That is, $v \left({\mathbf B}\right) = F$, so that:


 * $v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

Hence $\mathbf A$ and $\mathbf B$ are also equivalent in the sense of definition 2.

Definition 2 implies Definition 3
Let $\mathbf A, \mathbf B$ be equivalent according to definition 2.

By definition of the boolean interpretation of $\iff$:


 * $v \left({\mathbf A \iff \mathbf B}\right)= T$ iff $v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

Therefore, by hypothesis and definition of tautology:


 * $\mathbf A \iff \mathbf B$

is a tautology.

Definition 3 implies Definition 1
Let $\mathbf A, \mathbf B$ be equivalent according to definition 3.

That is, let $\mathbf A \iff \mathbf B$ be a tautology.

From the boolean interpretation of $\iff$, we have:


 * $v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

for every boolean interpretation $v$.

Therefore it immediately follows that:


 * $v \left({\mathbf A}\right) = T$ iff $v \left({\mathbf B}\right) = T$

i.e. $\mathbf A$ and $\mathbf B$ are equivalent in the sense of definition 1.