1. A -> (B & C)
2. C -> (E -> I)
3. ~(D -> I)
/ therefore, (A & G) -> (~I & ~E)
4. (A -> B) & (A -> C) 1, composition
5. A -> C 4, simplification
6. A -> (E -> I) 2,5, hypothetical syllogism
7. (A & E) -> I 6, exportation
8. ~I -> ~(A & E) 7, contrapositive
9. ~(~D v I) 3, material implication
10. ~~D & ~I 9, DeMorgan's law
11. D & ~I 10, double negation
12. ~I 11, simplification
13. ~(A & E) 8,12, hypothetical syllogism
14. ~A v ~E 13, DeMorgan's law
15. A -> ~E 14, material implication
16. (A -> ~E) v ~G 15, addition
17. (~A v ~E) v ~G 16, material implication
18. ~A v (~E v ~G) 17, association
19. ~A v (~G v ~E) 18, commutation
20. (~A v ~G) v ~E 19, association
21. ~(A & G) v ~E 20, DeMorgan's law
22. (A & G) -> ~E 21, exportation
23. ~E -> ~E & ~I 12, natural deduction
24. (A & G) -> (~E & ~I) 22,23, hypothetical syllogism
Edwin
