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Theorem euabsn2 4644
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 eu6 2660 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 absn 4566 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
32exbii 1849 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
41, 3bitr4i 281 1 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536   = wceq 1538  wex 1781  ∃!weu 2654  {cab 2802  {csn 4548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-sn 4549
This theorem is referenced by:  euabsn  4645  reusn  4646  absneu  4647  uniintab  4897  eusvobj2  7133  euabsneu  43477  aiotaexb  43503
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