SOLUTION: D ⊃ (F • S) / (B ⊃ D) ⊃ (B ⊃ S) I want to use a conditional proof to prove this answer using the given premise. I don't know where to start. Should I assume D?

Question 1159126: D ⊃ (F • S) / (B ⊃ D) ⊃ (B ⊃ S)
I want to use a conditional proof to prove this answer using the given premise. I don't know where to start. Should I assume D?
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!

     D  ⊃ (F • S)       / (B ⊃ D) ⊃ (B ⊃ S)

This one is a little different.  We first find a conditional statement which is
equivalent to the conclusion.  Then we do a conditional proof on the equivalent
statement and then the conclusion will follow.

Exportation says that p ⊃ (q ⊃ r) and (p • q) ⊃ r are equivalent.  Let's
substitute (B ⊃ D) for p, B for q, and S for r.  Then we have this

 (B ⊃ D) ⊃ (B ⊃ S)  <=> [(B ⊃ D) • B] ⊃ S

Now we know what to assume, which is the left side of the equivalent
statement to the conclusion: (B ⊃ D) • B

 1.  D ⊃ (F • S)        / (B ⊃ D) ⊃ (B ⊃ S) 

                        | 2. (B ⊃ D) • B    Assumption for conditional proof
                        | 3. B ⊃ D          2, simplification
                        | 4. B • (B ⊃ D)    2, commutation
                        | 5. B              4, simplification
                        | 6. D              3,5, modus ponens
                        | 7. F • S          1,6, modus ponens
                        | 8. S • F          7, commutation
                        | 9. S              8, simplification
10. [(B ⊃ D) • B] ⊃ S   lines 2--9  by conditional proof
11. (B ⊃ D) ⊃ (B ⊃ S)   10, exportation

Edwin

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