Using inference rules, prove the argument a^ (b→c) ^ [(a^b) →(d ∨¬c)] ^ b →d

please help! ( please try to explain when answering the question)
thanks a lot!

below is what i tried to do:
1. a
2. (b→c)
3. [(a^b) →(d ∨¬c)]
4. b
5. c (using modus ponens on 2, 4)
6. a^b ( conjunction using 1, 2)
7.

7. [¬(a ∧ b) ∨ (d ∨ ¬c)] (Implication Rule)
8. [(¬a ∨ ¬b) ∨ (d ∨ ¬c)] (DeMorgan's Law)
9. [¬a ∨ ¬b ∨ ¬c ∨ d] (removing some parentheses and rearranging terms)
10. Since we have (a,b,c) True, this becomes: [F ∨ d] = d