Theorem abid2 2287

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 170	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
2	1	abbi2i 2281	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
3	2	eqcomi 2169	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   = wceq 1343   ∈ wcel 2136  {cab 2151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161

This theorem is referenced by:  csbid  3053  abss  3211  ssab  3212  abssi  3217  notab  3392  inrab2  3395  dfrab2  3397  dfrab3  3398  notrab  3399  eusn  3650  dfopg  3756  iunid  3921  csbexga  4110  imai  4960  dffv4g  5483  frec0g  6365  dfixp  6666  euen1b  6769  acfun  7163  ccfunen  7205