Theorem abid2f 3014

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, 17-Nov-2019.)

Hypothesis

Ref	Expression
abid2f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
abid2f	⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Proof of Theorem abid2f

Step	Hyp	Ref	Expression
1		nfab1 2981	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
2		abid2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	cleqf 3012	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴))
4		abid 2805	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)
5	3, 4	mpgbir 1800	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114  {cab 2801  Ⅎwnfc 2963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965

This theorem is referenced by:  mptctf  30455  rabexgf  41288  ssabf  41373  abssf  41385