Theorem abid2f 2401

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 2377	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
3	1, 2	cleqf 2400	. . . 4 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}))
4		abid 2219	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)
5	4	bibi2i 227	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
6	5	albii 1519	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
7	3, 6	bitri 184	. . 3 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8		biid 171	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
9	7, 8	mpgbir 1502	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
10	9	eqcomi 2235	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2202  {cab 2217  Ⅎwnfc 2362

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364

This theorem is referenced by: (None)