The trick to proving the conclusion in this argument is to spot that M, on line 3, can be exported to make the consequent of a conditional (which itself is a conditional with M as the antecedent and N as the consequent). This one step allows us to set up a hypothetical syllogism with the proposition on line 4. From then on, it's child's play. If we take another path, we're in for a lot of hard work.
1. G ⊃ (H • I) 2. J ⊃ (H • K) 3. [(L ⊃ ~ G) • M] ⊃ N 4. (M ⊃ N) ⊃ (L • J) ∴ I v K 5. (L ⊃ ~ G) ⊃ (M ⊃ N) 6. (L ⊃ ~ G) ⊃ (L • J) 7. ~ (L ⊃ ~ G) v (L • J) 8. ~ (~ L v ~ G) v (L • J) 9. (L • G) v (L • J) 10. [(L • G) v L] • [(L • G) v J] 11. (L v L) • (G v L) • (L v J) • (G v J) 12. (G v J) • (L v L) • (G v L) • (L v J) 13. G v J 14. [G ⊃ (H • I)] • [J ⊃ (H • K)] 15. (H • I) v (H • K) 16. [(H • I) v H] • [(H • I) v K] 17. (H v H) • (I v H) • (H v K) • (I v K) 18. (I v K) • (H v H) • (I v H) • (H v K) 19. I v K |
3 Exp 4,5 HS 6 Impl 7 Impl 8 DM 9 Dist 10 Dist 11 Com 12 Simp 1,2 Conj 13,14 CD 15 Dist 16 Dist 17 Com 18 Simp |
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