This one is a bit of a doozy. Feel free to ask about any piece of the proof.
1. B > [(O v ~O) > (T v U)]
2. U > ~(G v ~G) / B > T
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3. ~(B > T) AIP
4. ~(~B v T) 3 MI
5. ~~B & ~T 4 DM
6. B & ~T 5 DN
7. ~T & B 6 Comm
8. B 6 Simp
9. ~T 7 Simp
10. (O v ~O) > (T v U) 1,8 MP
11. ~(O v ~O) v (T v U) 10 MI
12. (~O & ~~O) v (T v U) 11 DM
13. (~O & O) v (T v U) 12 DN
14. (T v U) v (~O & O) 13 Comm
15. ((T v U) v ~O) & ((T v U) v O) 14 Dist
16. ((T v U) v O) & ((T v U) v ~O) 15 Comm
17. (T v U) v O 16 Simp
18. T v (U v O) 17 Assoc
19. U v O 18,9 DS
20. ~~U v O 19 DN
21. ~U > O 20 MI
22. (T v U) v ~O 15 Simp
23. T v (U v ~O) 22 Assoc
24. U v ~O 23,9 DS
25. ~O v U 24 Comm
26. O > U 25 MI
27. ~U > U 21,26 HS
28. ~~U v U 27 MI
29. U v U 28 DN
30. U 29 Taut
31. ~(G v ~G) 2,30 MP
32. ~G & ~~G 31 DM
33. ~G & G 32 DN
34. B > T 3-33 IP
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Notes: To make things short and fit, I used the following abbreviations
AIP: Assumption for indirect proof. This is where you assume the complete opposite of the conclusion, and you try to find a contradiction. If a contradiction arises because you assumed that the opposite of the conclusion is true, then the conclusion must be true.
MI: Material Implication. This says that P > Q is the same as ~P v Q
DM: De Morgan's Law/Theorem: ~(P v Q) is the same as ~P & ~Q. Similarly, ~(P & Q) is the same as ~P v ~Q
DN: Double Negation: ~~P is the same as P
Comm: Commutation: P v Q is the same as Q v P. Also, P & Q is the same as Q & P
Assoc: Association: P v (Q v R) is the same as (P v Q) v R. In addition, P & (Q & R) is the same as (P & Q) & R
Simp: Simplification: P & Q can be simplified to just P (note: they are not equivalent)
Dist: Distribution: P v (Q & R) can be expressed as (P v Q) & (P v R). Also, P & (Q v R) can be written as (P & Q) v (P & R)
DS: Disjunctive Syllogism: If you have the statement P v Q and you know that ~P is the case, then you automatically can deduce Q (ex: I state that "I either had eggs or toast for breakfast" and I add in "I did NOT have eggs". So you can conclude that I must have had toast for breakfast)
HS: Hypothetical Syllogism: If you had P > Q and you have Q > R, then you can say that P > R. Ex: If it rains, then it is wet outside. If it is wet outside, then my grass will grow. Therefore, if it rains, then my grass will grow.
MP: Modus Ponens: If you have the conditional P > Q and you have P as a premise, then Q is the conclusion. Ex: If it rains, then it is wet outside. It rains. Therefore, it is wet outside.
Taut: Tautology: The basic idea that P v P is the same as P (ex: "I ate pizza or I ate pizza" is the same as "I ate pizza"). Similarly, P & P is the same as P as well (using the same reasoning)
IP: Indirect proof: This is the end of the proof by contradiction. If you start by assuming the opposite of the conclusion, do a bit of derivation, and you find a contradiction, then you have shown that the original conclusion must be valid. This is added to wrap things up and show that this conclusion was proven to be valid indirectly.