Theorem eusv1 4860

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 2053	. . . 4 |- ( A. x y = A -> y = A )
2		sp 2053	. . . 4 |- ( A. x z = A -> z = A )
3		eqtr3 2643	. . . 4 |- ( ( y = A /\ z = A ) -> y = z )
4	1, 2, 3	syl2an 494	. . 3 |- ( ( A. x y = A /\ A. x z = A ) -> y = z )
5	4	gen2 1723	. 2 |- A. y A. z ( ( A. x y = A /\ A. x z = A ) -> y = z )
6		eqeq1 2626	. . . 4 |- ( y = z -> ( y = A <-> z = A ) )
7	6	albidv 1849	. . 3 |- ( y = z -> ( A. x y = A <-> A. x z = A ) )
8	7	eu4 2518	. 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 954	1 |- ( E! y A. x y = A <-> E. y A. x y = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-cleq 2615

This theorem is referenced by:  eusvnfb  4862