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Theorem abid2f 2260
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 2237 . . . . 5 𝑥{𝑥𝑥𝐴}
31, 2cleqf 2259 . . . 4 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}))
4 abid 2083 . . . . . 6 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
54bibi2i 226 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ (𝑥𝐴𝑥𝐴))
65albii 1411 . . . 4 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝑥𝐴}) ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
73, 6bitri 183 . . 3 (𝐴 = {𝑥𝑥𝐴} ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
8 biid 170 . . 3 (𝑥𝐴𝑥𝐴)
97, 8mpgbir 1394 . 2 𝐴 = {𝑥𝑥𝐴}
109eqcomi 2099 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1294   = wceq 1296  wcel 1445  {cab 2081  wnfc 2222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224
This theorem is referenced by: (None)
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