Ring is Commutative iff Opposite Ring is Itself

Theorem
Let $\struct {R, +, \times}$ be a ring.

Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$.

Then $\struct {R, +, \times}$ is a commutative ring $\struct {R, +, \times} = \struct {R, +, *}$.

Proof
By definition of the opposite ring:
 * $\forall x, y \in R: x * y = y \times x$.

Necessary Condition
Let $\struct {R, +, \times}$ be a commutative ring.

Then:

Sufficient Condition
Let $\struct {R, +, \times} = \struct {R, +, *}$.

Then:

Thus by definition $\struct {R, +, \times}$ is a commutative ring.