MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusn Structured version   Visualization version   GIF version

Theorem reusn 4732
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 4730 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
2 df-reu 3374 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 df-rab 3430 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eqeq1i 2733 . . 3 ({𝑥𝐴𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
54exbii 1843 . 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 395   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2558  {cab 2705  ∃!wreu 3371  {crab 3429  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-reu 3374  df-rab 3430  df-sn 4630
This theorem is referenced by:  reuen1  9049  cshwrepswhash1  17071  frcond3  30078  vdgn1frgrv2  30105  ddemeas  33855
  Copyright terms: Public domain W3C validator