Theorem eusv2i 4433

Description: Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
eusv2i	|- ( E! y A. x y = A -> E! y E. x y = A )

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem eusv2i

Step	Hyp	Ref	Expression
1		nfeu1 2025	. . 3 |- F/ y E! y A. x y = A
2		nfcvd 2309	. . . . . 6 |- ( E! y A. x y = A -> F/_ x y )
3		eusvnf 4431	. . . . . 6 |- ( E! y A. x y = A -> F/_ x A )
4	2, 3	nfeqd 2323	. . . . 5 |- ( E! y A. x y = A -> F/ x y = A )
5		nf2 1656	. . . . 5 |- ( F/ x y = A <-> ( E. x y = A -> A. x y = A ) )
6	4, 5	sylib 121	. . . 4 |- ( E! y A. x y = A -> ( E. x y = A -> A. x y = A ) )
7		19.2 1626	. . . 4 |- ( A. x y = A -> E. x y = A )
8	6, 7	impbid1 141	. . 3 |- ( E! y A. x y = A -> ( E. x y = A <-> A. x y = A ) )
9	1, 8	eubid 2021	. 2 |- ( E! y A. x y = A -> ( E! y E. x y = A <-> E! y A. x y = A ) )
10	9	ibir 176	1 |- ( E! y A. x y = A -> E! y E. x y = A )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   F/wnf 1448   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-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  eusv2nf  4434