Theorem abid2 2350

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 2344	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
3	2	eqcomi 2233	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   = wceq 1395   ∈ wcel 2200  {cab 2215

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225

This theorem is referenced by:  csbid  3132  abss  3293  ssab  3294  abssi  3299  notab  3474  inrab2  3477  dfrab2  3479  dfrab3  3480  notrab  3481  eusn  3740  dfopg  3854  iunid  4020  csbexga  4211  imai  5083  dffv4g  5623  frec0g  6541  dfixp  6845  euen1b  6953  acfun  7385  ccfunen  7446