Theorem eusv2i 4449

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 2035	. . 3 |- F/ y E! y A. x y = A
2		nfcvd 2318	. . . . . 6 |- ( E! y A. x y = A -> F/_ x y )
3		eusvnf 4447	. . . . . 6 |- ( E! y A. x y = A -> F/_ x A )
4	2, 3	nfeqd 2332	. . . . 5 |- ( E! y A. x y = A -> F/ x y = A )
5		nf2 1666	. . . . 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 1636	. . . 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 2031	. 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 1351   = wceq 1353   F/wnf 1458   E.wex 1490   E!weu 2024

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-sbc 2961  df-csb 3056

This theorem is referenced by:  eusv2nf  4450