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Theorem euabsn2 3506
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
euabsn2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem euabsn2
StepHypRef Expression
1 df-eu 1951 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abeq1 2197 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 3458 . . . . . 6 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43bibi2i 225 . . . . 5 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1404 . . . 4 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 182 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
76exbii 1541 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
81, 7bitr4i 185 1 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948  {cab 2074  {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-v 2621  df-sn 3447
This theorem is referenced by:  euabsn  3507  reusn  3508  absneu  3509  uniintabim  3720  euabex  4043  nfvres  5321  eusvobj2  5620
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