Theorem eusv1 4437

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 1504	. . . 4 |- ( A. x y = A -> y = A )
2		sp 1504	. . . 4 |- ( A. x z = A -> z = A )
3		eqtr3 2190	. . . 4 |- ( ( y = A /\ z = A ) -> y = z )
4	1, 2, 3	syl2an 287	. . 3 |- ( ( A. x y = A /\ A. x z = A ) -> y = z )
5	4	gen2 1443	. 2 |- A. y A. z ( ( A. x y = A /\ A. x z = A ) -> y = z )
6		eqeq1 2177	. . . 4 |- ( y = z -> ( y = A <-> z = A ) )
7	6	albidv 1817	. . 3 |- ( y = z -> ( A. x y = A <-> A. x z = A ) )
8	7	eu4 2081	. 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 936	1 |- ( E! y A. x y = A <-> E. y A. x y = A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485   E!weu 2019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-cleq 2163

This theorem is referenced by:  eusvnfb  4439