Theorem abid2fOLD 2953

Description: Obsolete version of abid2f 2952 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.)

Hypothesis

Ref	Expression
abid2f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
abid2fOLD	⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Proof of Theorem abid2fOLD

Step	Hyp	Ref	Expression
1		nfab1 2924	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
2		abid2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	cleqf 2950	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴))
4		abid 2742	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)
5	3, 4	mpgbir 1831	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {cab 2738  Ⅎwnfc 2907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909

This theorem is used by: (None)