I'm asked to derive the validity of Universal Modus Tollens from the validity of Universal Instantiation and Modus Tollens. I'm new to this deriving/proving stuff, so I'm not sure if I'm doing it right, but here's what I came up with:
Universal instantiation says that if (1) is true:
(1) ∀x, P(x) → Q(x)
Then (2) is true for any particular item y
(2) P(y) → Q(y)
Modus Tollens says that if (2) is true and (3) is true:
Then (4) is true:
Therefore if (1) is true, and (3) is true, then (4) is true. In other words, the following argument is valid:
∀x, P(x) → Q(x) ~Q(y) ∴ ~P(y)
And that's Universal Modus Tollens.
Am I doing this right? Am I making any unwarranted assumptions or unsupported claims? Am I skipping any steps?