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Theorem reusn 3508
Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
reusn (∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem reusn
StepHypRef Expression
1 euabsn2 3506 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
2 df-reu 2366 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 df-rab 2368 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eqeq1i 2095 . . 3 ({𝑥𝐴𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
54exbii 1541 . 2 (∃𝑦{𝑥𝐴𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
61, 2, 53bitr4i 210 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948  {cab 2074  ∃!wreu 2361  {crab 2363  {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-reu 2366  df-rab 2368  df-v 2621  df-sn 3447
This theorem is referenced by:  reuen1  6498
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