SOLUTION: Use natural deduction to derive the conclusion of the following arguments. Do not use conditional proof or indirect proof. 1. A &#8835; (B &#8835; D) 2. A &#8835; (C &#8835; F)

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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):
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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

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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

SOLUTION: Use natural deduction to derive the conclusion of the following arguments. Do not use conditional proof or indirect proof. 1. A &#8835; (B &#8835; D) 2. A &#8835; (C &#8835; F)

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)