Class Equality is Transitive

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Theorem

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

Let $=$ denote class equality.


Then

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


Proof

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By Universal Generalisation and Biconditional is Transitive:

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


$\blacksquare$


Sources