I am working on this proof and wanted to know if I am using the ID natural deduction rule correctly. Can I just assume B and A based on that rule?
A ->(B->C)⊢ B->(A->C)
1. A -> (B->C) (given)
2. B (ID)
3. A (ID)
4. B -> C (->E 1,3)
5. C (->E 4,2)
6. A -> C (->I 3,5)
7. B->(A->C) (->I 2,6)
QED
I think you've used "ID" correctly (though I haven't seen the rule called "ID" before, and I don't know what it says exactly).
That said, when you used ->I you didn't just use steps 3 and 5 as your proof analysis suggests. You used all of the steps from 3 to 5. So, you'll want to notate that differently than you did... probably >I 3-5 for step 6. and >I 2-6 for step 7. Seeing that you used all of the steps here might come as clearer in an axiomatic context where you can prove the Deduction Meta-Theorem.