Theorem eusv1 4150

Description: Two ways to express single-valuedness of a class expression A(x). (Contributed by NM, 14-Oct-2010.)

Assertion

Ref	Expression
eusv1	|- (E!yA.x y = A <-> E.yA.x y = A)

Distinct variable groups:   x,y   y,A

Allowed substitution hint:   A(x)

Proof of Theorem eusv1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sp 1398	. . . 4 |- (A.x y = A -> y = A)
2		sp 1398	. . . 4 |- (A.x z = A -> z = A)
3		eqtr3 2056	. . . 4 |- ((y = A /\ z = A) -> y = z)
4	1, 2, 3	syl2an 273	. . 3 |- ((A.x y = A /\ A.x z = A) -> y = z)
5	4	gen2 1336	. 2 |- A.yA.z((A.x y = A /\ A.x z = A) -> y = z)
6		eqeq1 2043	. . . 4 |- (y = z -> (y = A <-> z = A))
7	6	albidv 1702	. . 3 |- (y = z -> (A.x y = A <-> A.x z = A))
8	7	eu4 1959	. 2 |- (E!yA.x y = A <-> (E.yA.x y = A /\ A.yA.z((A.x y = A /\ A.x z = A) -> y = z)))
9	5, 8	mpbiran2 847	1 |- (E!yA.x y = A <-> E.yA.x y = A)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378  E!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030

This theorem is referenced by:  eusvnfb  4152