How is the problem not going from 6 to 7? The false argument you gave uses PA + ((◻C)->C)|- C instead of PA |- ?((◻C)->C) when invoking the deduction theorem. In other words, the false argument is exactly the same except when it uses step 2 (which is Löb's Hypothesis). More specifically, making L doesn't depend on Löb's Hypothesis, and steps 1-6 don't depend on it either (as in they are still true in PA without the Hypothesis). Furthermore, assuming step 7 (ie PA + whatever you add |- step 7) step 8 must follow by A1.
How is the problem not going from 6 to 7? The false argument you gave uses PA + ((◻C)->C)|- C instead of PA |- ?((◻C)->C) when invoking the deduction theorem. In other words, the false argument is exactly the same except when it uses step 2 (which is Löb's Hypothesis). More specifically, making L doesn't depend on Löb's Hypothesis, and steps 1-6 don't depend on it either (as in they are still true in PA without the Hypothesis). Furthermore, assuming step 7 (ie PA + whatever you add |- step 7) step 8 must follow by A1.