Theorem eueq 1912

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

Step	Hyp	Ref	Expression
1		eqtr3t 1491	. . . 4 |- ((x = A /\ y = A) -> x = y)
2	1	gen2 981	. . 3 |- A.xA.y((x = A /\ y = A) -> x = y)
3	2	biantru 723	. 2 |- (E.x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
4		isset 1810	. 2 |- (A e. V <-> E.x x = A)
5		eqeq1 1478	. . 3 |- (x = y -> (x = A <-> y = A))
6	5	eu4 1408	. 2 |- (E!x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
7	3, 4, 6	3bitr4 183	1 |- (A e. V <-> E!x x = A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  E!weu 1378  Vcvv 1807

This theorem is referenced by:  eueq1 1913  moeq 1916  0ex 2706  snex 2745  euuni 2876  reuhyp 2900  fnopab2g 3608  fvopab2 3782  elrnopabg 3791  fopab2 3814  en2d 4387

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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