Equivalences are Interderivable

Context
Natural deduction.

Theorem
If two statement forms are interderivable, they are equivalent:

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

Proof
First, we show that if $$p \vdash q$$ and $$q \vdash p$$, then $$p \iff q$$:

Next, we show that if $$p \iff q$$, then $$p \vdash q$$ and $$q \vdash p$$:

Similarly:

Q.E.D.