Prove whether ¬∃xP(x) logically implies ¬∀xP(x) or not.

Question

So my logic is

¬∃xP(x)⟺ ∀x¬P(x)

and if ∀x¬P(x) is taken to be true, then for all x P(x) is false

and

¬∀xP(x)⟺ ∃x¬P(x)

which means that for some $x$, $P(x)$ is false

and if all $x$ is false, then that implies that some $x$ is false.

Is this correct?

Answer 1

You appear to be trying a semantic argument, rather than a syntactic derivation. Have you been taught any rules of inference, or are you at the stage of justifying the meaning for the symbols?

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$\forall x~\lnot P(x)$ says every entity denies the predicate.

$\exists x~P(x)$ says some entity affirms the predicate.

If every entity denies the predicate, then we deny that some entity affirms the predicate. $$\forall x~\lnot P(x)\implies \lnot\exists x~P(x)$$

If no entity affirms the predicate, then every entity denies the predicate. $$\lnot \exists x~P(x)\implies\forall x~\lnot P(x)$$