Theorem eusvnfb 4383

Description: Two ways to say that A ( x ) is a set expression that does not depend on x. (Contributed by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
eusvnfb	|- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem eusvnfb

Step	Hyp	Ref	Expression
1		eusvnf 4382	. . 3 |- ( E! y A. x y = A -> F/_ x A )
2		euex 2030	. . . 4 |- ( E! y A. x y = A -> E. y A. x y = A )
3		id 19	. . . . . . 7 |- ( y = A -> y = A )
4		vex 2692	. . . . . . 7 |- y e. _V
5	3, 4	eqeltrrdi 2232	. . . . . 6 |- ( y = A -> A e. _V )
6	5	sps 1518	. . . . 5 |- ( A. x y = A -> A e. _V )
7	6	exlimiv 1578	. . . 4 |- ( E. y A. x y = A -> A e. _V )
8	2, 7	syl 14	. . 3 |- ( E! y A. x y = A -> A e. _V )
9	1, 8	jca 304	. 2 |- ( E! y A. x y = A -> ( F/_ x A /\ A e. _V ) )
10		isset 2695	. . . . 5 |- ( A e. _V <-> E. y y = A )
11		nfcvd 2283	. . . . . . . 8 |- ( F/_ x A -> F/_ x y )
12		id 19	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
13	11, 12	nfeqd 2297	. . . . . . 7 |- ( F/_ x A -> F/ x y = A )
14	13	nfrd 1501	. . . . . 6 |- ( F/_ x A -> ( y = A -> A. x y = A ) )
15	14	eximdv 1853	. . . . 5 |- ( F/_ x A -> ( E. y y = A -> E. y A. x y = A ) )
16	10, 15	syl5bi 151	. . . 4 |- ( F/_ x A -> ( A e. _V -> E. y A. x y = A ) )
17	16	imp 123	. . 3 |- ( ( F/_ x A /\ A e. _V ) -> E. y A. x y = A )
18		eusv1 4381	. . 3 |- ( E! y A. x y = A <-> E. y A. x y = A )
19	17, 18	sylibr 133	. 2 |- ( ( F/_ x A /\ A e. _V ) -> E! y A. x y = A )
20	9, 19	impbii 125	1 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481   E!weu 2000   F/_wnfc 2269   _Vcvv 2689

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by:  eusv2nf  4385  eusv2  4386