Theorem abid2 2357

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 2349	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
3	2	eqcomi 2238	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   = wceq 1398   ∈ wcel 2205  {cab 2220

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230

This theorem is referenced by:  csbid  3149  abss  3311  ssab  3312  abssi  3317  notab  3495  inrab2  3498  dfrab2  3500  dfrab3  3501  notrab  3502  eusn  3770  dfopg  3886  iunid  4052  csbexga  4243  imai  5123  dffv4g  5672  frec0g  6641  dfixp  6948  euen1b  7056  modom2  7075  acfun  7527  ccfunen  7594  ballotfilem2  13172