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Theorem euabsn2 4230
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 2473 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abeq1 2730 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 4164 . . . . . 6 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43bibi2i 327 . . . . 5 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1744 . . . 4 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 264 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
76exbii 1771 . 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 1478   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  {cab 2607  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-sn 4149
This theorem is referenced by:  euabsn  4231  reusn  4232  absneu  4233  uniintab  4480  eusvobj2  6597
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