Theorem abid2f 2202

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 2180	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
3	1, 2	cleqf 2201	. . . 4 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}))
4		abid 2028	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑥 ∈ 𝐴)
5	4	bibi2i 216	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
6	5	albii 1359	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
7	3, 6	bitri 173	. . 3 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8		biid 160	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
9	7, 8	mpgbir 1342	. 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
10	9	eqcomi 2044	1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

Syntax hints:   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167

This theorem is referenced by: (None)