Definitions of exists unique 
$\exists!x_0 \in S,P(x_0)$
Definitions:
1.$\exists x_0 \in S, P(x_0)\wedge(\forall x_1,x_2 \in S, P(x_1)\wedge P(x_2)\rightarrow x_1=x_2)$
2.$\exists x_0 \in S, P(x_0)\wedge (\forall x_1 \in S,P(x_1)\rightarrow x_0=x_1)$
$3. \exists x_0 \in S, \ \forall x_1 \in S, (P(x_1) \leftrightarrow x_0 = x_1))$


Question: I saw people sometimes use first one and sometimes use the second one in uniqueness proofs, are they equivalent, if so, is it possible to prove it?

Passing from (1) to (2) is trivial, it suffices to let $x_2=x_0$. Now we will start with assuming (2) as true. We have
$$\forall x_1,x_2 \in S \big( P(x_1) \land P(x_2) \Rightarrow x_0=x_1 \land x_0=x_2\big)$$which leads to$$\forall x_1,x_2 \in S \big( P(x_1) \land P(x_2) \Rightarrow x_1=x_2\big)$$and we have just demonstrated (1).
