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Theorem abid2f 2199
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 {xx A} = A

Proof of Theorem abid2f
StepHypRef Expression
1 abid2f.1 . . . . 5 xA
2 nfab1 2177 . . . . 5 x{xx A}
31, 2cleqf 2198 . . . 4 (A = {xx A} ↔ x(x Ax {xx A}))
4 abid 2025 . . . . . 6 (x {xx A} ↔ x A)
54bibi2i 216 . . . . 5 ((x Ax {xx A}) ↔ (x Ax A))
65albii 1356 . . . 4 (x(x Ax {xx A}) ↔ x(x Ax A))
73, 6bitri 173 . . 3 (A = {xx A} ↔ x(x Ax A))
8 biid 160 . . 3 (x Ax A)
97, 8mpgbir 1339 . 2 A = {xx A}
109eqcomi 2041 1 {xx A} = A
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164
This theorem is referenced by: (None)
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