Theorem eusv2i 4187

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 1911	. . 3 |- F/ y E! y A. x y = A
2		nfcvd 2179	. . . . . 6 |- ( E! y A. x y = A -> F/_ x y )
3		eusvnf 4185	. . . . . 6 |- ( E! y A. x y = A -> F/_ x A )
4	2, 3	nfeqd 2192	. . . . 5 |- ( E! y A. x y = A -> F/ x y = A )
5		nf2 1558	. . . . 5 |- ( F/ x y = A <-> ( E. x y = A -> A. x y = A ) )
6	4, 5	sylib 127	. . . 4 |- ( E! y A. x y = A -> ( E. x y = A -> A. x y = A ) )
7		19.2 1529	. . . 4 |- ( A. x y = A -> E. x y = A )
8	6, 7	impbid1 130	. . 3 |- ( E! y A. x y = A -> ( E. x y = A <-> A. x y = A ) )
9	1, 8	eubid 1907	. 2 |- ( E! y A. x y = A -> ( E! y E. x y = A <-> E! y A. x y = A ) )
10	9	ibir 166	1 |- ( E! y A. x y = A -> E! y E. x y = A )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   F/wnf 1349   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-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  eusv2nf  4188