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Can anyone provide me with a reference which contains the proof of the validity of the following two implications?

  1. [ ∃x p(x) -> ∃x q(x) ] -> [ ∃x ( p(x) -> q(x) ) ]

  2. [ ∀ x ( p(x) -> q(x) ) ] -> [ ∀x p(x) -> ∀x q(x) ]

I tired to find a reference but I couldn't, most books provide that the universal and existential quantifiers don't distribute over the implication logical operator, but the books concluded that the above implications are valid without their proofs.

Is there any available book reference?

thanks

1 Answers 1

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I don't have any reference, but the two implications are easily explained:

  1. Assume $\exists p(x) \to \exists q(x)$. Now, if we assume that there is nothing that is a $p$, then for any object that I pick, the conditional $p(x) \to q(x)$ will be true, since $p(x)$ is false. So there is something such that this conditional is true. On the other hand, if there is soething that is a $p$, then by the assumption here is something tha is a $q$. And for such a something that is a $q$, the conditional $p(x) \to q(x)$ will then be true as well. So, either way, there is something for which the conditional is true, i.e. $\exists (p(x) \to q(x))$.

  2. Assume $\forall x(p(x) \to q(x))$. Now assume $\forall x p(x)$. Now take any arbitrary object. By $\forall x p(x)$ this object must be a $p$, and by $\forall x (p(x) \to q(x))$ that means the object must be a $q$. but since the object was arbitrary, that means all objects are $q$, i.e. $\forall x q(x)$. So, by conditional proof: $\forall x p(x) \to \forall x q(x)$