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Theorem abid2f 2307
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 2284 . . . . 5 𝑥{𝑥𝑥𝐴}
31, 2cleqf 2306 . . . 4 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}))
4 abid 2128 . . . . . 6 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
54bibi2i 226 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ (𝑥𝐴𝑥𝐴))
65albii 1447 . . . 4 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
73, 6bitri 183 . . 3 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
8 biid 170 . . 3 (𝑥𝐴𝑥𝐴)
97, 8mpgbir 1430 . 2 𝐴 = {𝑥𝑥𝐴}
109eqcomi 2144 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1330   = wceq 1332  wcel 1481  {cab 2126  wnfc 2269
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271
This theorem is referenced by: (None)
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