Theorem eusv1 4430

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 1499	. . . 4 |- ( A. x y = A -> y = A )
2		sp 1499	. . . 4 |- ( A. x z = A -> z = A )
3		eqtr3 2185	. . . 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 1438	. 2 |- A. y A. z ( ( A. x y = A /\ A. x z = A ) -> y = z )
6		eqeq1 2172	. . . 4 |- ( y = z -> ( y = A <-> z = A ) )
7	6	albidv 1812	. . 3 |- ( y = z -> ( A. x y = A <-> A. x z = A ) )
8	7	eu4 2076	. 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 931	1 |- ( E! y A. x y = A <-> E. y A. x y = A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   E!weu 2014

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-cleq 2158

This theorem is referenced by:  eusvnfb  4432