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Theorem reusn 4232
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 4230 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
2 df-reu 2914 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 df-rab 2916 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eqeq1i 2626 . . 3 ({𝑥𝐴𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
54exbii 1771 . 2 (∃𝑦{𝑥𝐴𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
61, 2, 53bitr4i 292 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  {cab 2607  ∃!wreu 2909  {crab 2911  {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-reu 2914  df-rab 2916  df-v 3188  df-sn 4149
This theorem is referenced by:  reuen1  7969  cshwrepswhash1  15733  frcond3  26998  vdgn1frgrv2  27024  ddemeas  30080
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