Theorem zfrep3cl

Description: An inference based on the Axiom of Replacement. Typically, ph defines a function from x to y . (Contributed by NM, 26-Nov-1995)

	Ref	Expression
Hypotheses	zfrep3cl.1	$$⊢A\in \mathrm{V}$$
	zfrep3cl.2	$$⊢x\in A\to \exists z\forall y\left(\phi \to y=z\right)$$
Assertion	zfrep3cl	$$⊢\exists z\forall y\left(y\in z\leftrightarrow \exists x\left(x\in A\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		zfrep3cl.1	$$⊢A\in \mathrm{V}$$
2		zfrep3cl.2	$$⊢x\in A\to \exists z\forall y\left(\phi \to y=z\right)$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
4	3 1 2	zfrepclf	$$⊢\exists z\forall y\left(y\in z\leftrightarrow \exists x\left(x\in A\wedge \phi \right)\right)$$