a biconditional
Some rules work both ways at once. 'A switch is on if and only if the bulb is lit' tells you two things in one breath: light follows the switch, and the switch follows the light. That two-way 'if and only if' is a biconditional.
A biconditional joins a statement p and a statement q as 'p if and only if q', often shortened to 'p iff q'. It asserts that the conditional 'if p then q' and its converse 'if q then p' are both true. As a result, p and q always have the same truth value: whenever one holds the other holds, and whenever one fails the other fails. To prove a biconditional you must prove both directions separately — first that p implies q, then that q implies p. The biconditional is true exactly when p and q match (both true or both false) and false when they differ.
Biconditionals are the natural form of a good definition. When we say 'a triangle is equilateral if and only if all three sides are equal', the two sides of the iff are interchangeable: each can stand for the other. This is why most definitions are secretly biconditionals — they give a condition that is both necessary and sufficient. Take care not to confuse a biconditional with a plain conditional: 'a number is divisible by 4 if it is divisible by 8' is a one-way conditional, not an iff, because the converse fails (12 is divisible by 4 but not by 8).
'An angle is a right angle if and only if it measures 90°' is a true biconditional: being a right angle forces 90°, and measuring 90° forces being a right angle. By contrast 'an angle is acute if and only if it is less than a right angle' must be checked both ways before you trust the iff.
An iff packs a conditional and its converse into one statement — prove both directions.
An 'if and only if' demands both directions; proving only 'if p then q' leaves the biconditional half-proved. Most definitions are biconditionals in disguise.