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Metamath Proof Explorer

Theorem abid2f

Description: A simplification of class abstraction. Theorem 5.2 of Quine p. 35. (Contributed by NM, 5-Sep-2011) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 26-Feb-2025)

Ref Expression
Hypothesis abid2f.1 ⊢ Ⅎ _ x A
Assertion abid2f ⊢ x | x ∈ A = A

Proof

Step Hyp Ref Expression
1 abid2f.1 ⊢ Ⅎ _ x A
2 1 eqabf ⊢ A = x | x ∈ A ↔ ∀ x x ∈ A ↔ x ∈ A
3 biid ⊢ x ∈ A ↔ x ∈ A
4 2 3 mpgbir ⊢ A = x | x ∈ A
5 4 eqcomi ⊢ x | x ∈ A = A