Theorem abid2 2471

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 2465	. 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  3144  csbexg  3147  abss  3336  ssab  3337  abssi  3342  notab  3526  inrab2  3529  dfrab2  3531  dfrab3  3532  notrab  3533  eusn  3797  uniintsn  3964  iunid  4022  imai  5011  epini  5022  clos1basesuc  5883  nnnc  6147