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 ∈ V
	zfrep3cl.2	⊢ ( 𝑥 ∈ 𝐴 → ∃ 𝑧 ∀ 𝑦 ( 𝜑 → 𝑦 = 𝑧 ) )
Assertion	zfrep3cl	⊢ ∃ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		zfrep3cl.1	⊢ 𝐴 ∈ V
2		zfrep3cl.2	⊢ ( 𝑥 ∈ 𝐴 → ∃ 𝑧 ∀ 𝑦 ( 𝜑 → 𝑦 = 𝑧 ) )
3		nfcv	⊢ Ⅎ 𝑥 𝐴
4	3 1 2	zfrepclf	⊢ ∃ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )