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Metamath Proof Explorer

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