Theorem eusv2i 4596

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 2097	. . 3 |- F/ y E! y A. x y = A
2		nfcvd 2393	. . . . . 6 |- ( E! y A. x y = A -> F/_ x y )
3		eusvnf 4594	. . . . . 6 |- ( E! y A. x y = A -> F/_ x A )
4	2, 3	nfeqd 2407	. . . . 5 |- ( E! y A. x y = A -> F/ x y = A )
5		nf2 1720	. . . . 5 |- ( F/ x y = A <-> ( E. x y = A -> A. x y = A ) )
6	4, 5	sylib 122	. . . 4 |- ( E! y A. x y = A -> ( E. x y = A -> A. x y = A ) )
7		19.2 1691	. . . 4 |- ( A. x y = A -> E. x y = A )
8	6, 7	impbid1 142	. . 3 |- ( E! y A. x y = A -> ( E. x y = A <-> A. x y = A ) )
9	1, 8	eubid 2093	. 2 |- ( E! y A. x y = A -> ( E! y E. x y = A <-> E! y A. x y = A ) )
10	9	ibir 177	1 |- ( E! y A. x y = A -> E! y E. x y = A )

Syntax hints:   -> wi 4   A.wal 1400   = wceq 1402   F/wnf 1513   E.wex 1545   E!weu 2086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is referenced by:  eusv2nf  4597