Theorem abid2fOLD 2937

Description: Obsolete version of abid2f 2936 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 2907	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
2		abid2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	cleqf 2934	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴))
4		abid 2718	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)
5	3, 4	mpgbir 1798	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   ↔ wb 206   = wceq 1539   ∈ wcel 2108  {cab 2714  Ⅎwnfc 2890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892

This theorem is referenced by: (None)