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Theorem reusn 3793
 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 (∃!x A φy{x A φ} = {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 3791 . 2 (∃!x(x A φ) ↔ y{x (x A φ)} = {y})
2 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
3 df-rab 2623 . . . 4 {x A φ} = {x (x A φ)}
43eqeq1i 2360 . . 3 ({x A φ} = {y} ↔ {x (x A φ)} = {y})
54exbii 1582 . 2 (y{x A φ} = {y} ↔ y{x (x A φ)} = {y})
61, 2, 53bitr4i 268 1 (∃!x A φy{x A φ} = {y})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  ∃!wreu 2616  {crab 2618  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-reu 2621  df-rab 2623  df-sn 3741 This theorem is referenced by: (None)
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