Theorem eusv2i 4440

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 2030	. . 3 |- F/ y E! y A. x y = A
2		nfcvd 2313	. . . . . 6 |- ( E! y A. x y = A -> F/_ x y )
3		eusvnf 4438	. . . . . 6 |- ( E! y A. x y = A -> F/_ x A )
4	2, 3	nfeqd 2327	. . . . 5 |- ( E! y A. x y = A -> F/ x y = A )
5		nf2 1661	. . . . 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 1631	. . . 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 2026	. 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 1346   = wceq 1348   F/wnf 1453   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-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  eusv2nf  4441