SOLUTION: I need help with proofs. Thank you! D →(B→C), B & −C ├ −D

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Question 978498: I need help with proofs. Thank you!
D →(B→C), B & −C ├ −D
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You can do a proof by contradiction.

Step 1) Assume the opposite of the conclusion. Assume ~~D which is equivalent to D

Step 2) Since we have D, we can use D -> (B -> C) and modus ponens to get B -> C

Step 3) Use simplification to go from B & ~C to just B. You can also free up ~C as well

Step 4) Use B and B -> C to get C (modus ponens again)

Step 5) The ~C and C contradict one another. So the original assumption is not possible. Therefore, the opposite of the assumption (D) is the only possibility.

So that's one way we can prove that ~D is the conclusion.