Theorem eusvnfb 4252

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 4251	. . 3 |- ( E! y A. x y = A -> F/_ x A )
2		euex 1975	. . . 4 |- ( E! y A. x y = A -> E. y A. x y = A )
3		id 19	. . . . . . 7 |- ( y = A -> y = A )
4		vex 2618	. . . . . . 7 |- y e. _V
5	3, 4	syl6eqelr 2176	. . . . . 6 |- ( y = A -> A e. _V )
6	5	sps 1473	. . . . 5 |- ( A. x y = A -> A e. _V )
7	6	exlimiv 1532	. . . 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 2619	. . . . 5 |- ( A e. _V <-> E. y y = A )
11		nfcvd 2226	. . . . . . . 8 |- ( F/_ x A -> F/_ x y )
12		id 19	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
13	11, 12	nfeqd 2239	. . . . . . 7 |- ( F/_ x A -> F/ x y = A )
14	13	nfrd 1456	. . . . . 6 |- ( F/_ x A -> ( y = A -> A. x y = A ) )
15	14	eximdv 1805	. . . . 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 4250	. . 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 1285   = wceq 1287   E.wex 1424   e. wcel 1436   E!weu 1945   F/_wnfc 2212   _Vcvv 2615

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830  df-csb 2923

This theorem is referenced by:  eusv2nf  4254  eusv2  4255