Equality of Ordered Pairs/Sufficient Condition

Theorem
Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs.

Let $a = c$ and $b = d$.

Then:
 * $\tuple {a, b} = \tuple {c, d}$

Proof
Suppose $a = c$ and $b = d$.

Then:
 * $\set a = \set c$

and:
 * $\set {a, b} = \set {c, d}$

Thus:
 * $\set {\set a, \set {a, b} } = \set {\set c, \set {c, d} }$

and so by the Kuratowski formalization:
 * $\tuple {a, b} = \tuple {c, d}$