1. Set up the vOut Strategy
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
-Bv-N GOAL
A v B
A>G
B>G G
A v B G
Set up the vOut Strategy

2. ... by making the missing conditionals new goals.
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
-Bv-N GOAL
A v B
A>G
B>G G
A v B G
... by making the missing conditionals new goals.

3. Prove the top goal using the >I Strategy
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
V>(-Bv-N) GOAL
-V>(-Bv-N) GOAL
-Bv-N 1,?,? vO
A v B
A>G
B>G G
A v B G
Prove the top goal using the >I Strategy

4. Apply >O whenever you can.
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
4. V PA
-Bv-N GOAL
V>(-Bv-N) 4-? >I
-V>(-Bv-N) GOAL
-Bv-N 1,?,? vO
A v B
A>G
B>G G
A v B G
Apply >O whenever you can.

5. ... and your goal comes by vIn
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
4. V PA
-B 2,4 >O
-Bv-N GOAL
V>(-Bv-N) 4-? >I
-V>(-Bv-N) GOAL
-Bv-N 1,?,? vO
A v B
A>G
B>G G
A v B G
... and your goal comes by vIn

6. Now use >In to obtain your second goal.
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
4. V PA
5. -B 2,4 >O
6. -Bv-N 5 vI
7. V>(-Bv-N) 4-6 >I
-V>(-Bv-N) GOAL
-Bv-N 1,7,? vO
A v B
A>G
B>G G
A v B G
Now use >In to obtain your second goal.

7. Apply >O whenever you can.
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
4. V PA
5. -B 2,4 >O
6. -Bv-N 5 vI
7. V>(-Bv-N) 4-6 >I
8. -V PA
-Bv-N GOAL
-V>(-Bv-N) 8-? >I
-Bv-N 1,7,? vO
A v B
A>G
B>G G
A v B G
Apply >O whenever you can.

8. ... and your goal comes by vIn
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
4. V PA
5. -B 2,4 >O
6. -Bv-N 5 vI
7. V>(-Bv-N) 4-6 >I
8. -V PA
-N 3,8 >O
-Bv-N GOAL
-V>(-Bv-N) 8-? >I
-Bv-N 1,7,? vO
A v B
A>G
B>G G
A v B G
... and your goal comes by vIn

9. The proof is now complete.
Proofs Using vOut
1. V v -V A
2. V>-B A
3. -V>-N A
4. V PA
5. -B 2,4 >O
6. -Bv-N 5 vI
7. V>(-Bv-N) 4-6 >I
8. -V PA
9. -N 3,8 >O
10. -Bv-N 9 vI
11. -V>(-Bv-N) 8-10>I
12. -Bv-N 1,7,11 vO
A v B
A>G
B>G G
A v B G
The proof is now complete.

10. Do &Out whenever you can
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
(A&B)v(A&C) GOAL
Do &Out whenever you can

11. Set up the vOut Strategy
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
(A&B)v(A&C) GOAL
Set up the vOut Strategy

12. Set the missing conditionals as new goals.
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
(A&B)v(A&C) 3,?,? vO
Set the missing conditionals as new goals.

13. Use >In to prove the first goal.
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
B >[(A&B)v(A&C)] GOAL
C >[(A&B)v(A&C)] GOAL
(A&B)v(A&C) 3,?,? vO
Use >In to prove the first goal.

14. Prove A&B so that you can use vIn to obtain the goal.
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
4. B PA
(A&B)v(A&C) GOAL
B >[(A&B)v(A&C)] 4-? >I
C >[(A&B)v(A&C)] GOAL
(A&B)v(A&C) 3,7,? vO
Prove A&B so that you can use vIn to obtain the goal.

15. Now the subproof can be completed with vIn.
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
4. B PA
5. A&B 2,4 &I
(A&B)v(A&C) GOAL
B >[(A&B)v(A&C)] 4-? >I
C >[(A&B)v(A&C)] GOAL
(A&B)v(A&C)
Now the subproof can be completed with vIn.

16. Now set up the >In Strategy to prove the second goal.
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
4. B PA
5. A&B 2,4 &I
6. (A&B)v(A&C) 5 vI
7. B >[(A&B)v(A&C)] 4-6 >I
C >[(A&B)v(A&C)] GOAL
(A&B)v(A&C) 3,7,? vO
Now set up the >In Strategy to prove the second goal.

17. This subproof can be completed in the same way.
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
4. B PA
5. A&B 2,4 &I
6. (A&B)v(A&C) 5 vI
7. B >[(A&B)v(A&C)] 4-6 >I
8. C PA
(A&B)v(A&C) GOAL
C >[(A&B)v(A&C)] 8-? >I
(A&B)v(A&C) 3,7,? vO
This subproof can be completed in the same way.

18. The proof is now complete.
Proofs Using vOut
A v B G
A v B
A>G
B>G G
1. A&(BvC) A
2. A 1 &O
3. B v C 1 &O
4. B PA
5. A&B 2,4 &I
6. (A&B)v(A&C) 5 vI
7. B >[(A&B)v(A&C)] 4-6 >I
8. C PA
9. A&C 2,8 &I
10. (A&B)v(A&C) 9 vI
11. C >[(A&B)v(A&C)] 8-10 >I
12. (A&B)v(A&C) 3,7,11 vO
The proof is now complete.