How do you solve a proof given ¬A ∨ ¬(¬B ∧ (¬A ∨ B)) without any premises?
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Negate the conclusion and demonstrate a contradiction:
{1} 1. ¬(¬A ∨ ¬(¬B ∧ (¬A ∨ B))) Assum.
{2} 2. ¬A Assum.
{2} 3. ¬A ∨ ¬(¬B ∧ (¬A ∨ B)) 2 ∨I
{1,2} 4. ⊥ 1,3 ∧I
{1} 5. ¬¬A 2,4 RAA
{1} 6. A 5 DNE
{7} 7. ¬(¬B ∧ (¬A ∨ B)) Assum.
{7} 8. ¬A ∨ ¬(¬B ∧ (¬A ∨ B)) 7 ∨I
{1,7} 9. ⊥ 1,8 ∧I
{1} 10. ¬¬(¬B ∧ (¬A ∨ B)) 7,9 RAA
{1} 11. ¬B ∧ (¬A ∨ B) 10 DNE
{1} 12. ¬B 11 ∧E
{1} 13. ¬A ∨ B 11 ∧E
{14} 14. ¬A Assum. (13 1st Disj.)
{1,14} 15. ⊥ 6,14 ∧I (13 1st Concl.)
{16} 16. B Assum. (13 2nd Disj.)
{1,16} 17. ⊥ 12,16 ∧I
{1} 18. ⊥ 13,14,15,16,17 EE
- 19. ¬A ∨ ¬(¬B ∧ (¬A ∨ B)) 1,18 RAA