Question 1171443: Regular Proof:
Cv(~B*~A)
~(~E*D) / (B>C)*(D>E)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this proof step-by-step, using standard propositional logic rules.
**Given Premises:**
1. ~(~E * D)
2. (B ⊃ C) * (D ⊃ E)
**Goal:**
Prove C v (~B * ~A)
**Proof:**
1. ~(~E * D) (Premise)
2. (B ⊃ C) * (D ⊃ E) (Premise)
3. ~(~E) v ~D (1, De Morgan's Law)
4. E v ~D (3, Double Negation)
5. B ⊃ C (2, Simplification)
6. D ⊃ E (2, Simplification)
7. ~D ⊃ E (6, Implication)
8. E v ~D (4, repeat for clarity)
9. ~E ⊃ ~~D (7, Contraposition)
10. ~E ⊃ D (9, Double Negation)
11. E v ~D (4, repeat)
12. ~E v D (10, implication)
13. E v ~D, ~E v D (4,12, repeat)
14. ~D v E (11, Commutation)
15. ~D ⊃ E (14, Implication)
16. ~E v D, ~D v E (12, 15, repeat)
17. ~B v C (5, Implication)
18. C v ~B (17, Commutation)
19. C v (~B * ~A) (18, Addition)
**Explanation:**
1. **Premise 1:** We are given the negation of a conjunction.
2. **Premise 2:** We have a conjunction of two implications.
3. **Step 3:** We apply De Morgan's Law to premise 1, distributing the negation.
4. **Step 4:** We apply Double Negation to the first term in step 3.
5. **Step 5:** We simplify premise 2, extracting the first implication.
6. **Step 6:** We simplify premise 2, extracting the second implication.
7. **Step 7:** We rewrite step 6 in disjunctive form.
8.-16: repeat steps for clarity.
9. **Step 17:** We rewrite step 5 in disjunctive form.
10. **Step 18:** We apply Commutation to step 17.
11. **Step 19:** We apply Addition to step 18, adding `~A`.
**Therefore, we have proven C v (~B * ~A).**
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