Theorem eueq 1427

Description: Equality has existential uniqueness.

Assertion

Ref	Expression
eueq	|- (A e. V <-> E!x x = A)

Distinct variable group(s):   x,A

Proof of Theorem eueq

Step	Hyp	Ref	Expression
1		cleq2 1110	. . . . 5 |- (y = A -> (x = y <-> x = A))
2	1	biimparc 327	. . . 4 |- ((x = A /\ y = A) -> x = y)
3	2	gen2 681	. . 3 |- A.xA.y((x = A /\ y = A) -> x = y)
4	3	biantru 543	. 2 |- (E.x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
5		isset 1351	. 2 |- (A e. V <-> E.x x = A)
6		cleq1 1107	. . 3 |- (x = y -> (x = A <-> y = A))
7	6	eu4 1036	. 2 |- (E!x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
8	4, 5, 7	3bitr4 158	1 |- (A e. V <-> E!x x = A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E!weu 1007   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  eueq1 1428  moeq 1431  reuhyp 1581  euuni 1954  fvopab2 2878  fopab2 2891  en2d 3303

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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