Without classical negation, the absence of a proof of φ does not constructively yield a proof of ¬φ, so the biconditional holds only under classical assumptions the claim silently imports.

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Biconditional(in formal logic): A logical statement that says two things are true if and only if each other is true; it's a two-way relationship (like saying 'you can vote if and only if you're 18').
Classical assumptions(what constructive logic challenges): The traditional rules of logic that have been used for centuries, which assume that every statement is either completely true or completely false.
Classical negation(contrasted with constructive logic): The traditional logical rule that says if something is not true, then its opposite must be true—like how something is either raining or not raining, with no middle ground.
Constructively(describing how proof works in constructive logic): In a way that requires actually building or demonstrating something with concrete steps, rather than just assuming it's true based on logical rules alone.

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