Can you prove or disprove following equation with possible example

Question

\[\exists(x)P(x) \Leftrightarrow \exists(x) \bar P(x)\]

Comments

Take a variable x1 which have p(x) is true then its !P(x) will false, thus

it is true <-> false.

Hence above is not true.

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Let P(x) = x eats meat.

∃(x)P(x) means some people eat meat. Can't we say that  from set of all people ..if some people eat meat then some people do not eat meat.Please advise where I am wrong.

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∃(x)P(x) means some people eat meat

First if you want to apply this predicate to set of people then you should use Domain of Discourse as people. Because if Domain is not mentioned then this statement is applied over all entity of the universe.

Ref:-https://en.m.wikipedia.org/wiki/Domain_of_discourse

Refer the example section.

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Ok is it true if we define domain of discourse as set of all living people?