{1} | (1) (W∨P)→I | Premise |
{2} | (2) I→C∨S | Premise |
{3} | (3) S→U | Premise |
{4} | (4) ¬C∧¬U | Premise |
{4} | (5) ¬U∧¬C | ∧-commutativity on 4 |
{4} | (6) ¬C | ∧-elimination on 4 |
{4} | (7) ¬U | ∧-elimination on 5 |
{3} | (8) ¬U→¬S | contrapositive of 3 |
{3,4} | (9) ¬S | →-elimination on 7 and 8 |
{2} | (10) ¬(C∨S)→¬I | contrapositive of 2 |
{1} | (11) ¬I→¬(W∨P) | contrapositive of 1 |
{1,2} | (12) ¬(C∨S)→¬(W∨P) | syllogism on 10 and 11 |
{3,4} | (13) ¬C∧¬S | ∧-introduction on 6 and 9 |
{3,4} | (14) ¬(C∨S) | DeMorgan's law on 13 |
{1,2,3,4} | (15) ¬(W∨P) | →-elimination on 12 and 14 |
{1,2,3,4} | (16) ¬W∧¬P | DeMorgan's law on 15 |
{1,2,3,4} | (17) ¬W | ∧-elimination on 16 |
{1} | (1) D∨¬L | Premise |
{2} | (2) M→¬D | Premise |
{1} | (3) ¬L∨D | ∨-commutativity on 1 |
{1} | (4) L→D | equivalence of implication and disjunction on 3 |
{2} | (5) ¬(¬D)→¬M | contrapositive of 2 |
{6} | (6) D | Premise |
{6} | (7) ¬(¬D) | double negation of 6 |
{2,6} | (8) ¬M | →-elmination on 5 and 7 |
{2} | (9) D→¬M | rule of conditional proof on 6 and 8 |
{1,2} | (10) L→¬M | syllogism on 4 and 9 |
{1} | (1) J→¬N | Premise |
{2} | (2) ¬J→¬D | Premise |
{3} | (3) ¬D→A | Premise |
{4} | (4) R→N | Premise |
{1} | (5) ¬(¬N)→¬J | contrapositive of 1 |
{6} | (6) N | Premise |
{6} | (7) ¬(¬N) | double negative of 6 |
{1,6} | (8) ¬J | →-elimination on 5 and 7 |
{1} | (9) N→¬J | rule of conditional proof on 6 and 8 |
{1,4} | (10) R→¬J | syllogism on 4 and 9 |
{1,2,4} | (11) R→¬D | syllogism on 2 and 10 |
{1,2,3,4} | (12) R→A | syllogism on 3 and 11 |
{1,2,3,4} | (13) ¬R∨A | law of equivalence of implication and disjunction on 12 |
{1} | (1) S∨O | Premise |
{2} | (2) S→¬E | Premise |
{3} | (3) O→M | Premise |
{2} | (4) ¬(¬E)→¬S | contrapositive of 2 |
{5} | (5) E | Premise |
{5} | (6) ¬(¬E) | double negative of 5 |
{2,5} | (7) ¬S | →-elimination of 4 and 6 |
{2} | (8) E→¬S | conditional proof on 5 and 7 |
{9} | (9) ¬S | Premise |
{1,9} | (10) O | modus tollendo tollens on 1 and 9 |
{1} | (11) ¬S→O | conditional proof on 9 and 10 |
{1,2} | (12) E→O | syllogism on 8 and 11 |
{1,2,3} | (13) E→M | syllogism on 3 and 12 |
{1,2,3} | (14) ¬E∨M | equivalence of implication and disjunction on 13 |