Theorem abid2 2314

Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)

Assertion

Ref	Expression
abid2	⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem abid2

Step	Hyp	Ref	Expression
1		biid 171	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
2	1	abbi2i 2308	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
3	2	eqcomi 2197	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2164  {cab 2179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189

This theorem is referenced by:  csbid  3089  abss  3249  ssab  3250  abssi  3255  notab  3430  inrab2  3433  dfrab2  3435  dfrab3  3436  notrab  3437  eusn  3693  dfopg  3803  iunid  3969  csbexga  4158  imai  5022  dffv4g  5552  frec0g  6452  dfixp  6756  euen1b  6859  acfun  7269  ccfunen  7326