Theorem abid2 2470

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 227	. . 3 ⊢ (x ∈ A ↔ x ∈ A)
2	1	abbi2i 2464	. 2 ⊢ A = {x ∣ x ∈ A}
3	2	eqcomi 2357	1 ⊢ {x ∣ x ∈ A} = A

Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by:  csbid  3143  csbexg  3146  abss  3335  ssab  3336  abssi  3341  notab  3525  inrab2  3528  dfrab2  3530  dfrab3  3531  notrab  3532  eusn  3796  uniintsn  3963  iunid  4021  imai  5010  epini  5021  clos1basesuc  5882  nnnc  6146