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A Harder Example

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(PvQ)>R

-[(P>R)&(Q>R)]

(PvQ)>R | (P>R)&(Q>R)

(PvQ)>R

-[(P>R)&(Q>R)]

(PvQ)>R | (P>R)&(Q>R)

(-&) Rule

-(A&B)

-A -B

(PvQ)>R

-[(P>R)&(Q>R)]

(PvQ)>R | (P>R)&(Q>R)

1

(-&) Rule

-(A&B)

-A -B

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

1

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

1

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

(->) Rule

-(A>B)

A

-B

1

2

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

(->) Rule

-(A>B)

A

-B

P

-R

1

2

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

(->) Rule

-(A>B)

A

-B

P

-R

1

2

3

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

(->) Rule

-(A>B)

A

-B

P

-R

Q

-R

1

2

3

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

1

2

3

(>) Rule

A>B

-A B

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

1

2

3

(>) Rule

A>B

-A B

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

THE RESULT OF APPLYINGA RULE MUST APPEAR ONEVERY OPEN BRANCH BELOW THE PLACE IT IS APPLIED.

1

2

3

4

(>) Rule

A>B

-A B

-(PvQ) R

-(PvQ) R

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

THE RESULT OF APPLYINGA RULE MUST APPEAR ONEVERY OPEN BRANCH BELOW THE PLACE IT IS APPLIED.

1

2

3

4

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

1

2

3

4

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

1

2

3

4

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

(-v) Rule

-(AvB)

-A

-B

1

2

3

4

5

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

(-v) Rule

-(AvB)

-A

-B

-P

-Q

1

2

3

4

5

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

(-v) Rule

-(AvB)

-A

-B

-P

-Q

-(AvB) =

-A&-B

1

2

3

4

5

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

(-v) Rule

-(AvB)

-A

-B

-P

-Q

1

2

3

4

5

6

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

(-v) Rule

-(AvB)

-A

-B

-P

-Q

-P

-Q

1

2

3

4

5

6

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

P

-R

Q

-R

-P

-Q

*

-P

-Q

*

5

1

2

3

4

6

-(PvQ) R

*

-(PvQ) R

*

(PvQ)>R

-[(P>R)&(Q>R)]

-(P>R) -(Q>R)

(PvQ)>R | (P>R)&(Q>R)

VALID

P

-R

Q

-R

-P

-Q

*

-P

-Q

*

It is best to do non-branching steps first.

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