# Properties of Biconditional

## Theorem

### Rule of Material Equivalence

#### Formulation 1

$p \iff q \dashv \vdash \paren {p \implies q} \land \paren {q \implies p}$

#### Formulation 2

$\vdash \left({p \iff q}\right) \iff \left({\left({p \implies q}\right) \land \left({q \implies p}\right)}\right)$

### Biconditional as Disjunction of Conjunctions

#### Formulation 1

$p \iff q \dashv \vdash \paren {p \land q} \lor \paren {\neg p \land \neg q}$

#### Formulation 2

$\vdash \paren {p \iff q} \iff \paren {\paren {p \land q} \lor \paren {\neg p \land \neg q} }$

### Biconditional Equivalent to Biconditional of Negations

#### Formulation 1

$p \iff q \dashv \vdash \neg p \iff \neg q$

#### Formulation 2

$\vdash \left({p \iff q}\right) \iff \left({\neg p \iff \neg q}\right)$

### Biconditional iff Disjunction implies Conjunction

#### Formulation 1

$p \iff q \dashv \vdash \left({p \lor q}\right) \implies \left({p \land q}\right)$

#### Formulation 2

$\vdash \left({p \iff q}\right) \iff \left({\left({p \lor q}\right) \implies \left({p \land q}\right)}\right)$