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Why does “unless” mean “if not”?

Harry Gensler. [*Introduction to Logic*](https://www.amazon.com/Introduction-Logic-Harry-J-Gensler/dp/1138910597) (2017 3 ed). p 169.
~~>“Unless” is also equivalent to “if not”; so we also could use “$({\sim}B \supset D)$ (“If you~~
don’t breathe, then you’ll die”).”
Nicholas JJ Smith, [*Logic: The Laws of Truth*](https://www.amazon.com/Logic-Truth-Nicholas-J-J-Smith/dp/0691151636) (2012). p 115.
>The statement “P unless Q” means that if Q is not true, P is true—so we
translate it as $¬ \, Q→P$.
Using solely the original meaning of "unless" below, please expound why? How does definition 1 below ≡ if not? I know that definition 1 is obsolete, but I'm interested in the etymology. [*OED* Third Edition, June 2017](http://www.oed.com/view/Entry/215075). [Screenshot](https://i.stack.imgur.com/iZLrR.jpg).
> †**A.** adv. Only in conjunctional phrases followed by _than_ or _that_.
>
>1. Forming a conjunctional phrase introducing a case in which an exception to a preceding negative statement (expressed or implied) will or may exist: (not) on a less or lower condition, requirement, etc., _than_ (what is specified). _Obsolete_.

Harry Gensler. [*Introduction to Logic*](https://www.amazon.com/Introduction-Logic-Harry-J-Gensler/dp/1138910597) (2017 3 ed). p 169.
>“Unless” is also equivalent to “if not”; so we also could use “(∼B ⊃ D) (“If you
don’t breathe, then you’ll die”).”
Nicholas JJ Smith, [*Logic: The Laws of Truth*](https://www.amazon.com/Logic-Truth-Nicholas-J-J-Smith/dp/0691151636) (2012). p 115.
>The statement “P unless Q” means that if Q is not true, P is true—so we
translate it as $¬ \, Q→P$.
Using solely the original meaning of "unless" below, please expound why? How does definition 1 below ≡ if not? I know that definition 1 is obsolete, but I'm interested in the etymology. [*OED* Third Edition, June 2017](http://www.oed.com/view/Entry/215075). [Screenshot](https://i.stack.imgur.com/iZLrR.jpg).
> †**A.** adv. Only in conjunctional phrases followed by _than_ or _that_.
>
>1. Forming a conjunctional phrase introducing a case in which an exception to a preceding negative statement (expressed or implied) will or may exist: (not) on a less or lower condition, requirement, etc., _than_ (what is specified). _Obsolete_.

logic