Theorem eueq 1913

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 1492	. . . 4 |- ((x = A /\ y = A) -> x = y)
2	1	gen2 982	. . 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 1811	. 2 |- (A e. V <-> E.x x = A)
5		eqeq1 1479	. . 3 |- (x = y -> (x = A <-> y = A))
6	5	eu4 1409	. 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 953   = wceq 955   e. wcel 957  E.wex 979  E!weu 1379  Vcvv 1808

This theorem is referenced by:  eueq1 1914  moeq 1917  0ex 2707  snex 2746  euuni 2877  reuhyp 2901  fnopab2g 3612  fvopab2 3786  elrnopabg 3795  fopab2 3818  en2d 4390

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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