New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  eueq GIF version

Theorem eueq 3008
 Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eueq (A V ↔ ∃!x x = A)
Distinct variable group:   x,A

Proof of Theorem eueq
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2372 . . . 4 ((x = A y = A) → x = y)
21gen2 1547 . . 3 xy((x = A y = A) → x = y)
32biantru 491 . 2 (x x = A ↔ (x x = A xy((x = A y = A) → x = y)))
4 isset 2863 . 2 (A V ↔ x x = A)
5 eqeq1 2359 . . 3 (x = y → (x = Ay = A))
65eu4 2243 . 2 (∃!x x = A ↔ (x x = A xy((x = A y = A) → x = y)))
73, 4, 63bitr4i 268 1 (A V ↔ ∃!x x = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859 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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  eueq1  3009  moeq  3012  fnopab2g  5206
 Copyright terms: Public domain W3C validator