# Class Equality is Transitive

## Theorem

Let $A$, $B$, and $C$ be classes.

Let $=$ denote class equality.

Then

$\left({ A = B \land B = C }\right) \implies B = A$

## Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

$\forall x: \left({ \left({ x \in A \iff x \in B }\right) \land \left({ x \in B \iff x \in C }\right) }\right) \implies \forall x: \left({ x \in A \iff x \in C }\right)$ by Universal Generalisation and Biconditional is Transitive
$\blacksquare$