SOLUTION: Use natural deduction to derive the conclusion of the following arguments. Do not use conditional proof or indirect proof. 1. A ⊃ (B ⊃ D) 2. A ⊃ (C ⊃ F)

Question 1118993: Use natural deduction to derive the conclusion of the following arguments. Do not use conditional proof or indirect proof.
1. A ⊃ (B ⊃ D)
2. A ⊃ (C ⊃ F) / A ⊃ [(B ⊃ D) � (C ⊃ F)]
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
There is this logical equivalence, given in many tables where they include equivalences involving conditionals:
(A�>X) & (A�>Y) == A�>(X&Y)

So we can write

1.  A �> (B�>D)
2.  A �> (C�>F)
3.  (A�>(B�>D)) & (A�>(C�>F))        1,2 Conjunction (CONJ)
4.  A �> ((B �> D) & (C �> F))       3 Logical equivalence

��
Conditional Proof for comparison:

1.  A �> (B �> D)            Premise
2. A �> (C �> F)             Premise
3. :: A                      Conditional Proof assumption
4. :: B �> D                 3,1 MP
5. :: C �> F                 3,2 MP
6. :: (B �> D) & (C �> F)    4,5 CONJ
7. A�>((B �> D) & (C �> F))  3-6 Conditional Proof

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SOLUTION: Use natural deduction to derive the conclusion of the following arguments. Do not use conditional proof or indirect proof. 1. A ⊃ (B ⊃ D) 2. A ⊃ (C ⊃ F)