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

Question

Quizplus · Accepted Answer

Starting with the given formula, we can apply the following deduction:

(∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Qxy]

= (∀x)[~Px \lor (∃y)Qxy] ⊃ [~(∃x)Px \lor (∃x)(∃y)Qxy]

= (∀x)[~Px ⊃ ~(∃y)Qxy] ⊃ [(∀x)~Px \lor (∃x)(∃y)Qxy]

= (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [(∀x)~Px \lor (∃x)(∃y)Qxy]

= (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [(∀x)∼Px \lor (∃x)(∃y)Qxy]

Therefore, option B is derivable in F.

(∀X)[Px ⊃ (∃Y)Qxy] ⊃ [(∃X)Px ⊃ (∃X)(∃y)Qxy] -Which of the Following

(∀X)[Px ⊃ (∃Y)Qxy] ⊃ [(∃X)Px ⊃ (∃X)(∃y)Qxy]
-Which of the Following