I don't really like the way that proof is presented, and I know this isn't what you asked, but if you want to prove ~(P -> Q) is mutually implicated with P ^ ~Q, we can just compare the truth tables.
P implies Q has a certain truth table
| P | Q | P -> Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Which means ~ (P -> Q) ought to have the opposite truth table.
| P | Q | ~ (P -> Q) |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | F |
| F | F | F |
The only row on that truth table is the row where P is true and Q is false.
