Theorem eusv1 4184

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 1401	. . . 4 |- ( A. x y = A -> y = A )
2		sp 1401	. . . 4 |- ( A. x z = A -> z = A )
3		eqtr3 2059	. . . 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 1339	. 2 |- A. y A. z ( ( A. x y = A /\ A. x z = A ) -> y = z )
6		eqeq1 2046	. . . 4 |- ( y = z -> ( y = A <-> z = A ) )
7	6	albidv 1705	. . 3 |- ( y = z -> ( A. x y = A <-> A. x z = A ) )
8	7	eu4 1962	. 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 848	1 |- ( E! y A. x y = A <-> E. y A. x y = A )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   E!weu 1900

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-cleq 2033

This theorem is referenced by:  eusvnfb  4186