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The axiom is

$$\forall x \ \exists y \ \forall z \ (\varphi(z,x) \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ \varphi(y,x))$$

Consider

$$\forall x \ \exists y \ \forall z \ (z\in x \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ y\in x )$$

This is an instance of the axiom. Now the left side is true (take $y=x$). The right side expresses the existence of the union of $X$.

For another

The other version likeof the axiom:

$$\forall x\in X \ \exists y \ \varphi(x,y) \to \ \exists Y \ \forall x\in X \ \exists y\in Y \ \varphi(x,y)$$

you can use something like

$$\forall x \in P(X) \exists y \ (|x|=1 \to x=\{y\}) \to \exists Y \ \forall x\in P(X) \ \exists y\in Y \ (|x|=1 \to x=\{y\})$$

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The axiom is

$$\forall x \ \exists y \ \forall z \ (\varphi(z,x) \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ \varphi(y,x))$$

Consider

$$\forall x \ \exists y \ \forall z \ (z\in x \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ y\in x )$$

This is an instance of the axiom. Now the left side is true (take $y=x$). The right side expresses the existence of the union of $X$.

For another version like

$$\forall x\in X \ \exists y \ \varphi(x,y) \to \ \exists Y \ \forall x\in X \ \exists y\in Y \ \varphi(x,y)$$

you can use something like

$$\forall x \in P(X) \exists y \ (|x|=1 \to x=\{y\}) \to \exists Y \ \forall x\in P(X) \ \exists y\in Y \ (|x|=1 \to x=\{y\})$$

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$$\forall x \ \exists y \ \forall z \ (z\in x \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ y\in x )$$

This is an instance of the axiom. Now the left side is true (take $y=x$). The remaining part is right side expresses the existence of the union of $X$.

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