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Theorem euabsn2 3466
 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 1919 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abeq1 2163 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 3419 . . . . . 6 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43bibi2i 220 . . . . 5 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1375 . . . 4 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 177 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
76exbii 1512 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
81, 7bitr4i 180 1 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
 Colors of variables: wff set class Syntax hints:   ↔ wb 102  ∀wal 1257   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ∃!weu 1916  {cab 2042  {csn 3402 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-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sn 3408 This theorem is referenced by:  euabsn  3467  reusn  3468  absneu  3469  uniintabim  3679  euabex  3988  nfvres  5233  eusvobj2  5525
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