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Theorem abid2f 2218
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 2196 . . . . 5 𝑥{𝑥𝑥𝐴}
31, 2cleqf 2217 . . . 4 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}))
4 abid 2044 . . . . . 6 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
54bibi2i 220 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ (𝑥𝐴𝑥𝐴))
65albii 1375 . . . 4 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
73, 6bitri 177 . . 3 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
8 biid 164 . . 3 (𝑥𝐴𝑥𝐴)
97, 8mpgbir 1358 . 2 𝐴 = {𝑥𝑥𝐴}
109eqcomi 2060 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183
This theorem is referenced by: (None)
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