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Theorem abid2f 2283
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 𝑥𝐴
Assertion
Ref Expression
abid2f {𝑥𝑥𝐴} = 𝐴

Proof of Theorem abid2f
StepHypRef Expression
1 abid2f.1 . . . . 5 𝑥𝐴
2 nfab1 2260 . . . . 5 𝑥{𝑥𝑥𝐴}
31, 2cleqf 2282 . . . 4 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}))
4 abid 2105 . . . . . 6 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
54bibi2i 226 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ (𝑥𝐴𝑥𝐴))
65albii 1431 . . . 4 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
73, 6bitri 183 . . 3 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
8 biid 170 . . 3 (𝑥𝐴𝑥𝐴)
97, 8mpgbir 1414 . 2 𝐴 = {𝑥𝑥𝐴}
109eqcomi 2121 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1314   = wceq 1316  wcel 1465  {cab 2103  wnfc 2245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247
This theorem is referenced by: (None)
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