Theorem eusv1 4210

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

Assertion

Ref	Expression
eusv1	|- ( E! y A. x y = A <-> E. y A. 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 1442	. . . 4 |- ( A. x y = A -> y = A )
2		sp 1442	. . . 4 |- ( A. x z = A -> z = A )
3		eqtr3 2101	. . . 4 |- ( ( y = A /\ z = A ) -> y = z )
4	1, 2, 3	syl2an 283	. . 3 |- ( ( A. x y = A /\ A. x z = A ) -> y = z )
5	4	gen2 1380	. 2 |- A. y A. z ( ( A. x y = A /\ A. x z = A ) -> y = z )
6		eqeq1 2088	. . . 4 |- ( y = z -> ( y = A <-> z = A ) )
7	6	albidv 1746	. . 3 |- ( y = z -> ( A. x y = A <-> A. x z = A ) )
8	7	eu4 2004	. 2 |- ( E! y A. x y = A <-> ( E. y A. x y = A /\ A. y A. z ( ( A. x y = A /\ A. x z = A ) -> y = z ) ) )
9	5, 8	mpbiran2 883	1 |- ( E! y A. x y = A <-> E. y A. x y = A )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   = wceq 1285   E.wex 1422   E!weu 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-cleq 2075

This theorem is referenced by:  eusvnfb  4212