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Theorem euabsn2 4292
 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 2502 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abeq1 2762 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 4226 . . . . . 6 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43bibi2i 326 . . . . 5 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1787 . . . 4 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 264 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
76exbii 1814 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
81, 7bitr4i 267 1 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃!weu 2498  {cab 2637  {csn 4210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sn 4211 This theorem is referenced by:  euabsn  4293  reusn  4294  absneu  4295  uniintab  4547  eusvobj2  6683
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