Properties of Biconditional

Context
Natural deduction

Theorem

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


 * $$p \iff q \vdash \lnot p \iff \lnot q$$
 * $$\lnot p \iff \lnot q \vdash p \iff q$$

Proof
$$p \iff q \vdash \left({p \land q}\right) \lor \left({\lnot p \land \lnot q}\right)$$:

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

$$p \iff q \vdash \lnot p \iff \lnot q$$:

$$\lnot p \iff \lnot q \vdash p \iff q$$: