A(x) is a predicate logic formula. A is a property (predicate), x is a variable. ∃!A(x) would mean that exactly one x exists which has the property A.
First thing that comes up is: ∃x( A(x) ∧ ∀y( A(y) → (x=y) ),
where y is a variable, but my assignment forbids the use of functional symbols such as =.
How do I rewrite this without using any functional symbols? Only the use of →, ⇔, ¬, ∨, ∧, ∃, ∀, variables, formula A(x) and brackets are allowed.
My idea was to simply write this: ∃x( A(x) ∧ ∀y( A(y) → (A(x)⇔A(y)) )
not sure if it's correct, though.
Thank you in advance.
By the way, the question is not a duplicate, as I'm trying to rewrite this without the use of =.