Theorem eusvnfb 4204

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 4203	. . 3 |- ( E! y A. x y = A -> F/_ x A )
2		euex 1971	. . . 4 |- ( E! y A. x y = A -> E. y A. x y = A )
3		id 19	. . . . . . 7 |- ( y = A -> y = A )
4		vex 2604	. . . . . . 7 |- y e. _V
5	3, 4	syl6eqelr 2170	. . . . . 6 |- ( y = A -> A e. _V )
6	5	sps 1470	. . . . 5 |- ( A. x y = A -> A e. _V )
7	6	exlimiv 1529	. . . 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 300	. 2 |- ( E! y A. x y = A -> ( F/_ x A /\ A e. _V ) )
10		isset 2605	. . . . 5 |- ( A e. _V <-> E. y y = A )
11		nfcvd 2220	. . . . . . . 8 |- ( F/_ x A -> F/_ x y )
12		id 19	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
13	11, 12	nfeqd 2233	. . . . . . 7 |- ( F/_ x A -> F/ x y = A )
14	13	nfrd 1453	. . . . . 6 |- ( F/_ x A -> ( y = A -> A. x y = A ) )
15	14	eximdv 1801	. . . . 5 |- ( F/_ x A -> ( E. y y = A -> E. y A. x y = A ) )
16	10, 15	syl5bi 150	. . . 4 |- ( F/_ x A -> ( A e. _V -> E. y A. x y = A ) )
17	16	imp 122	. . 3 |- ( ( F/_ x A /\ A e. _V ) -> E. y A. x y = A )
18		eusv1 4202	. . 3 |- ( E! y A. x y = A <-> E. y A. x y = A )
19	17, 18	sylibr 132	. 2 |- ( ( F/_ x A /\ A e. _V ) -> E! y A. x y = A )
20	9, 19	impbii 124	1 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   F/_wnfc 2206   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  eusv2nf  4206  eusv2  4207