SOLUTION: Solve the following proof using natural deduction (rules of replacement and rules of implication). 1. D v Y 2. Y ⊃ ~(Z ⊃ D) / Y ≡ ~D

Algebra ->  Proofs -> SOLUTION: Solve the following proof using natural deduction (rules of replacement and rules of implication). 1. D v Y 2. Y ⊃ ~(Z ⊃ D) / Y ≡ ~D       Log On


   

Question 1029804: Solve the following proof using natural deduction (rules of replacement and rules of implication).
1. D v Y
2. Y ⊃ ~(Z ⊃ D)
/ Y ≡ ~D

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
1. D v Y --------------------Hypothesis
2. ~~D v Y --------------------double negation
3. ~D ⊃ Y ---------------------material implication
4. Y ⊃ ~(Z ⊃ D) ---------------hypothesis
5. Y ⊃ ~(~Z v D) --------------M.I.
6. Y ⊃ (Z & ~D) ---------------double neg. and deMorgan's law
7. ~Yv(Z & ~D) ----------------M.I.
8. (~Y v Z)&(~Y v ~D)-----------distributivity
9. ~Y v ~D -------------------simplification
10. Y ⊃ ~D -------------------M.I.
11. (~D ⊃ Y)&(Y ⊃ ~D) ---------conjunction on #3 and #10
12. Y ≡ ~D -------------------- #11 and logical equivalence for ≡