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Theorem abid2f 2184
 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 2162 . . . . 5 x{xx A}
31, 2cleqf 2183 . . . 4 (A = {xx A} ↔ x(x Ax {xx A}))
4 abid 2010 . . . . . 6 (x {xx A} ↔ x A)
54bibi2i 216 . . . . 5 ((x Ax {xx A}) ↔ (x Ax A))
65albii 1339 . . . 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 1322 . 2 A = {xx A}
109eqcomi 2026 1 {xx A} = A
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  {cab 2008  Ⅎwnfc 2147 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149 This theorem is referenced by: (None)
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