Theorem rabid2

Description: An "identity" law for restricted class abstraction. Prefer rabid2im if one direction is sufficient. (Contributed by NM, 9-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof shortened by Wolf Lammen, 24-Nov-2024)

	Ref	Expression
Assertion	rabid2	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfcv	⊢ Ⅎ 𝑥 𝐴
2	1	rabid2f	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )