Theorem abid2f 2515

Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypothesis

Ref	Expression
abid2f.1	⊢ ℲxA

Assertion

Ref	Expression
abid2f	⊢ {x ∣ x ∈ A} = A

Proof of Theorem abid2f

Step	Hyp	Ref	Expression
1		abid2f.1	. . . . 5 ⊢ ℲxA
2		nfab1 2492	. . . . 5 ⊢ Ⅎx{x ∣ x ∈ A}
3	1, 2	cleqf 2514	. . . 4 ⊢ (A = {x ∣ x ∈ A} ↔ ∀x(x ∈ A ↔ x ∈ {x ∣ x ∈ A}))
4		abid 2341	. . . . . 6 ⊢ (x ∈ {x ∣ x ∈ A} ↔ x ∈ A)
5	4	bibi2i 304	. . . . 5 ⊢ ((x ∈ A ↔ x ∈ {x ∣ x ∈ A}) ↔ (x ∈ A ↔ x ∈ A))
6	5	albii 1566	. . . 4 ⊢ (∀x(x ∈ A ↔ x ∈ {x ∣ x ∈ A}) ↔ ∀x(x ∈ A ↔ x ∈ A))
7	3, 6	bitri 240	. . 3 ⊢ (A = {x ∣ x ∈ A} ↔ ∀x(x ∈ A ↔ x ∈ A))
8		biid 227	. . 3 ⊢ (x ∈ A ↔ x ∈ A)
9	7, 8	mpgbir 1550	. 2 ⊢ A = {x ∣ x ∈ A}
10	9	eqcomi 2357	1 ⊢ {x ∣ x ∈ A} = A

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by: (None)