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Theorem reusn 4693
Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
reusn (∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem reusn
StepHypRef Expression
1 euabsn2 4691 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
2 df-reu 3357 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 df-rab 3411 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eqeq1i 2742 . . 3 ({𝑥𝐴𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
54exbii 1851 . 2 (∃𝑦{𝑥𝐴𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
61, 2, 53bitr4i 303 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2567  {cab 2714  ∃!wreu 3354  {crab 3410  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-reu 3357  df-rab 3411  df-sn 4592
This theorem is referenced by:  reuen1  8976  cshwrepswhash1  16982  frcond3  29255  vdgn1frgrv2  29282  ddemeas  32875
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