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Unique Existential Quantifier

I am given this unique existential quantifier: , which means there is only one single unique x in the domain that statisfy P(x)=true. But I don't understand how would this be equivalent?

Attachment 20955Wouldn't mean there is a possibility that there is more than one in the domain, so long as ? There could have been like 2 s that is different from all of the other s that satisfy P(x)=true. Then in this case, this isn't unique anymore right?

I also attached a diagram of how I thought of this, with the green zone being the and the white circles within the green zone being the . In the diagram, it exists more than one "x" and is not equals to "y", which makes me not understand the equivalence of to .

thanks!