(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px]
-Which of the following propositions is also derivable in F?
A) (∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)~Px ⊃ (∃x)(∃y)Rxy]
B) (∀x)[(∃y)Rxy ⊃ ~Px] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]
C) (∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]
D) (∀x)[~(∃y)Rxy ⊃ Px] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]
E) (∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)~Px ⊃ (∃x)(∃y)~Rxy]

Question

Quizplus · Accepted Answer

We can first simplify the main implication as follows: 
(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px] 
≡ (∀x)(∀y)[(Px ⊃ Rxy) ⊃ (∼Rxy ⊃ ∼Px)]

Then we apply the rule of contraposition to the antecedent of the consequent: 
(∀x)(∀y)[(Px ⊃ Rxy) ⊃ (Px ⊃ ~∼Px)] 
≡ (∀x)(∀y)[(Px ⊃ Rxy) ⊃ Px]

Finally, we use existential instantiation on x to obtain: 
(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]

(∀X)[Px ⊃ (∃Y)Rxy] ⊃ [(∀X)(∀y)∼Rxy ⊃ ∼(∃X)Px]