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Metamath Proof Explorer

Theorem abid2fOLD

Description: Obsolete version of abid2f as of 26-Feb-2025. (Contributed by NM, 5-Sep-2011) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 17-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis abid2f.1 ⊢ Ⅎ _ x A
Assertion abid2fOLD ⊢ x | x ∈ A = A

Proof

Step Hyp Ref Expression
1 abid2f.1 ⊢ Ⅎ _ x A
2 nfab1 ⊢ Ⅎ _ x x | x ∈ A
3 2 1 cleqf ⊢ x | x ∈ A = A ↔ ∀ x x ∈ x | x ∈ A ↔ x ∈ A
4 abid ⊢ x ∈ x | x ∈ A ↔ x ∈ A
5 3 4 mpgbir ⊢ x | x ∈ A = A