According to the binding spell, if any one of them dies the spell gets broken.
This is the conversation between Edward Coventry and the book owner:
Edward picks up the knife
Book Owner: Ah, ah, ah, ah! Remember the mirror incantation.
Book Owner: You hurt me, you're only hurting yourself.
Edward stabs himself
Book Owner: Sanetur! Reparetur! Restituatur!
Edward: Here's the thing about a mirror incantation. It only mimics the injuries. It doesn't guarantee certain death.
Edward: You never should have threatened my son.
Book Owner: You killed yourself... for petty revenge?
Edward: All I have to do is outlive you.
Book Owner: You want everything the Vade Maecum offers... but you'll never be able to pay the price.
Edward: You can die thinking that.
Book Owner: You will never be able to sacrifice your own son.
Book owner dies
Edward removes the knife and heals himself
The mirror incantation doesn't guarantee certain death, it only mimics the injuries... So both of them have the exact same injuries, but they might not die at the exact same time. Once he was sure the book owner was dead, the spell got broken, and he healed himself.