Properties of Biconditional

Theorem

 * $$p \iff q \dashv \vdash \left({p \and q}\right) \or \left({\neg p \and \neg q}\right)$$
 * $$p \iff q \dashv \vdash \neg p \iff \neg q$$
 * $$p \iff q \dashv \vdash \left({p \or q}\right) \implies \left({p \and q}\right)$$

Proof by Natural Deduction
By the tableau method:

The argument reverses:

The argument reverses:

Proof by Truth Table
Let $$v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula $$\phi$$ of two variables $$p, q$$.

We see that $$v \left({p \iff q}\right) = v \left({\left({p \and q}\right) \or \left({\neg p \and \neg q}\right)}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.

We see that $$v \left({p \iff q}\right) = v \left({\neg p \iff \neg q}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.

We see that $$v \left({p \iff q}\right) = v \left({\left({p \or q}\right) \implies \left({p \and q}\right)}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.