I am trying to convert the unique existential quantifier into plain logic. I have two statements with different placements of the 'for all' quantifier.
∃x ∀y, $(P(y) ∧ P(x)) \rightarrow (y=x)$
∃x, $(P(x) ∧ (∀y, P(y) \rightarrow y=x))$
I'm having trouble figuring out what the difference between these two statements is. The second one I found online and the first one I wrote myself.
If you could let me know the difference, I would greatly appreciate it.