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# State/Axiom of replacement

Axiom: (Axiom of Replacement) Suppose that $\Phi$ is a binary predicate. Then, $$(\forall x, \exists y, \Phi(x,y) ) \Rightarrow (\forall X, \exists Y, y \in Y \Leftrightarrow (\exists x \in X, \Phi(x,y) ) ).$$

Informally, given a set $X$, and binary predicate $\Phi$ satisfying a "functional" condition, one may replace each element $x \in X$ with the set $\{ y \vert \Phi(x,y) \}$ to obtain a new set $Y$.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Binary predicate

The following statements and definitions rely on the material of this page:

To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/ZFC axioms of set theory

This is one of the Zermelo-Fraenkel axioms of set theory