#38

(E → Q) ⋀ (E ⋁ B) ⋀ Q’→ B

1. E → Q hyp (simplification)

2. E ⋁ B hyp (simplification)

3. Q’ hyp (simplification)

4. E’ 1,3 Modus Tollens

5. B 2,4 Disjunctive syllogism

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1. E → Q hyp (simplification)

2. E ⋁ B hyp (simplification)

3. Q’ hyp (simplification)

4. E’⋁ Q 1 implication

5. Q ⋁ E’ 4 commutative

6. E’ 3,5 Disjunctive syllogism

7. B 2,6 Disjunctive syllogism

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1. E → Q hyp (simplification)

2. E ⋁ B hyp (simplification)

3. Q’ hyp (simplification)

4. Q’→ E’ 1 contraposition

5. E’ 4,3 Modus Ponens

6. (E’)’ ⋁ B 2 double negation

7. E’→ B 6 implication

8 B 7,5 Modus Ponens

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1. E → Q hyp (simplification)

2. E ⋁ B hyp (simplification)

3. Q’ hyp (simplification)

4. (E’)’ ⋁ B 2 double negation

5. E’→ B 4 implication

6. E’ ⋁ Q 1 implication

7. (E’ ⋁ Q) ⋀ Q’ 6,3 conjunction

8. Q’ ⋀ (E’ ⋁ Q) 7 commutative

9. (Q’ ⋀ E’) ⋁ (Q’ ⋀ Q) 8 distributive

10. (E’ ⋀ Q’) ⋁ 0 9 complement property

11. (E’ ⋀ Q’) 10 identity property

12. E’ 11 simplification

13. B 5,12 Modus Ponens

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