Equivalence of Definitions of Set Equality

Definition 1 implies Definition 2
Let $S = T$ by Definition 1.

Then:

Similarly:

Thus by the Rule of Conjunction:
 * $S \subseteq T \land T \subseteq S$

and so $S$ and $T$ are equal by Definition 2.

Definition 2 implies Definition 1
Let $S = T$ by Definition 2:


 * $S \subseteq T \land T \subseteq S$

First:

Then:

Thus by Biconditional Introduction:
 * $\forall x: \left({x \in S \iff x \in T}\right)$

and so $S$ and $T$ are equal by Definition 1.