The claim was that "all cryptographic codes that depend on the adversary having limited computing power would be broken."
Here's a (very slightly) more rigorous justification:
If P=NP, then any NP problem is in P with at most a polynomial slowdown. That is, if there's an algorithm taking T steps on a non-deterministic Turing machine, we can solve it on a deterministic Turing machine in f(T) steps, where f is a polynomial. Presumably, a "practical" algorithm would be one for which f has a low degree.
The kinds of algorithms we're concerned about in cryptography (and plenty of other fields) already have low time complexity. For example, generating or verifying an HMAC is O(n) in the length of the input. So if we had a way to solve NP problems with a low-degree polynomial slowdown, we could break HMACs in low-degree polynomial time.
It doesn't matter that there are O(n^1000) problems out there that would still be realistically unsolvable, because those problems don't have practical applications in the first place.
