Theorem eueq 1963

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		eqtr3 1541	. . . 4 |- ((x = A /\ y = A) -> x = y)
2	1	gen2 1024	. . 3 |- A.xA.y((x = A /\ y = A) -> x = y)
3	2	biantru 736	. 2 |- (E.x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
4		isset 1861	. 2 |- (A e. V <-> E.x x = A)
5		eqeq1 1528	. . 3 |- (x = y -> (x = A <-> y = A))
6	5	eu4 1452	. 2 |- (E!x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
7	3, 4, 6	3bitr4i 190	1 |- (A e. V <-> E!x x = A)

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995   = wceq 997   e. wcel 999  E.wex 1021  E!weu 1422  Vcvv 1858

This theorem is referenced by:  eueq1 1964  moeq 1967  0ex 2766  snex 2806  euuni 2938  reuhyp 2962  fnopab2g 3673  fvopab2 3848  elrnopabg 3857  fopab2 3880  en2d 4461

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-v 1859

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