Complete this proof in 19 lines . . (or less)?

Question

I need the following proof to be completed in as few lines as possible. Complete it from anywhere between 22-25 lines, 20-21 lines, or for the preferred 19 lines. We are only supposed to use the "Primitive Rules" (&out, arrow out, double arrow out, wedge out, &in, double arrow in, wedge in, and the box rules arrow in, dash in, and dash out) and the "Recommended Derived Rules" (Double Negation, DeMorgan's, Arrow Exchange, Modus Tollens, and Disjunctive Argument).

I am doing this on my own but I wanted to see what others would get to compare to my method - our prof. said to be creative!
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(H --> B)& -((RvS)vT)
(E&F)vH
(Sv-S)--> -(D-->E)



Conclusion: -(D--> -B)&H

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Thanks!

Answer 1

1. (H --> B) & -( (R v S) v T )
2. (E & F) v H
3. (S v -S) --> -(D --> E)

/ Therefore, -(D --> -B) & H

Proof:

4. H --> B { 1, Simplification }
5. - ( (R v S) v T ) { 1, Simplification }
6. -(R v S) & -T { 5, DeMorgan's Law }
7. -(R v S) { 6, Simplification }
8. -T { 6, Simplification }
9. -R & -S {7, DeMorgan's}
10. -S {9, Simplification}
11. -S v S {10, Addition}
12. S v -S. {11, commutation}
13. -(D --> E) {3, 12, Modus Ponens}
14. -(-D v E) {13, Material Implication)
15. -(-D) & -E {14, DeMorgan's Law}
16. D & -E. {15, Double Negation}
17. D. {16, Simplification}
18. -E. {16, Simplification}
. . .19. D --> -B {Assumption for an indirect proof.}
. . .20. -B. {17, 19, Modus Ponens}
. . .21. -H. {4, 20, Modus Tollens}
. . .22. E & F. {2, 21, Disjunctive Syllogism.}
. . .23. E. {22, Simplification}
. . .24. E & -E. {18, 23, Conjunction}
25. - (D --> -B) { 19-24, Indirect Proof }
. . .26. -H. {Assumption for an indirect proof.}
. . .27. E & F. {2, 26, Disjunctive Syllogism}
. . .28. E. {27, Simplification}
. . .29. E & -E {18, 28, Conjunction}
30. --H {26-29, Indirect Proof}
31. H. {30, Double Negation}
32. -(D --> -B) & H {25, 31, Conjunction}

Hmm... I can't seem to do this without resorting to conditional proofs and indirect proofs. Moreover, it appears this ended up being more lines than what you said. But this is the proof. :)

Answer 2

I don't recognize your rule names. In spite of teaching logic for 28 years. Logic texts vary much more than math in names and notation. I'll give you the drift, you'll have to fill in the reasons.

1st of all, in prem. 3, (Sv-S) is always true, and texts REALLY vary on how they let you pull tautologies out of thin air. Assuming you know how, from 3 you get
4. -(D→E)
5. -(-DvE) ........ 4, arrow out?
6. D&-E ........... 5 DeMorgan
7. D ................. 6
8. -E .................6
9. -Ev-F ............ 8
10. -(E&F) .........9 DeMorgan
11. H ................ 10, 2
12. H→B ...........1
13. B ..................12, 11
14. D&B ..............7, 13
15. -(-(D&B)) ........14 double n
16. -(-Dv-B) ..........15 DeMorgan
17. -(D→-B) ..........16
18. -(D→-B)&H ......17, 11

I'm curious what text you're using. Sounds like it's abandoned most of the old names for inference rules.