Equivalence of Definitions of Involutive Mapping

$(1)$ implies $(2)$
Let $f: A \to A$ be an involutive mapping by definition 1.

Then by definition:
 * $(1): \quad \forall x \in A: \map f {\map f x} = x$

Let $\map f x = y$.

Then by substituting $y$ for $\map f x$ into $(1)$:
 * $\forall x \in A: \map f y = x$

Thus $f: A \to A$ is an involutive mapping by definition 2.

$(2)$ implies $(1)$
Let $f: A \to A$ be an involutive mapping by definition 2.

Then by definition:
 * $\forall x, y \in A: \map f x = y \implies \map f y = x$

and so substituting $\map f x$ for $y$ we have:
 * $\map f {\map f x} = x$

Thus $f: A \to A$ is an involutive mapping by definition 1.

$(1)$ iff $(3)$
This is demonstrated in Mapping is Involution iff Bijective and Symmetric.