ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reusn GIF version

Theorem reusn 3678
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 3676 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
2 df-reu 2475 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 df-rab 2477 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eqeq1i 2197 . . 3 ({𝑥𝐴𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
54exbii 1616 . 2 (∃𝑦{𝑥𝐴𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦})
61, 2, 53bitr4i 212 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wex 1503  ∃!weu 2038  wcel 2160  {cab 2175  ∃!wreu 2470  {crab 2472  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-reu 2475  df-rab 2477  df-v 2754  df-sn 3613
This theorem is referenced by:  reuen1  6822
  Copyright terms: Public domain W3C validator