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Theorem eueq 1907
Description: Equality has existential uniqueness.
Assertion
Ref Expression
eueq |- (A e. V <-> E!x x = A)
Distinct variable group:   x,A
Proof of Theorem eueq
StepHypRef Expression
1 eqtr3t 1486 . . . 4 |- ((x = A /\ y = A) -> x = y)
21gen2 980 . . 3 |- A.xA.y((x = A /\ y = A) -> x = y)
32biantru 722 . 2 |- (E.x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
4 isset 1805 . 2 |- (A e. V <-> E.x x = A)
5 eqeq1 1473 . . 3 |- (x = y -> (x = A <-> y = A))
65eu4 1403 . 2 |- (E!x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
73, 4, 63bitr4 183 1 |- (A e. V <-> E!x x = A)
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  Vcvv 1802
This theorem is referenced by:  eueq1 1908  moeq 1911  0ex 2701  snex 2740  euuni 2871  reuhyp 2895  fnopab2g 3602  fvopab2 3776  elrnopabg 3785  fopab2 3808  en2d 4381
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803
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