Theorem eusvnfb 4186

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 4185	. . 3 |- ( E! y A. x y = A -> F/_ x A )
2		euex 1930	. . . 4 |- ( E! y A. x y = A -> E. y A. x y = A )
3		id 19	. . . . . . 7 |- ( y = A -> y = A )
4		vex 2560	. . . . . . 7 |- y e. _V
5	3, 4	syl6eqelr 2129	. . . . . 6 |- ( y = A -> A e. _V )
6	5	sps 1430	. . . . 5 |- ( A. x y = A -> A e. _V )
7	6	exlimiv 1489	. . . 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 290	. 2 |- ( E! y A. x y = A -> ( F/_ x A /\ A e. _V ) )
10		isset 2561	. . . . 5 |- ( A e. _V <-> E. y y = A )
11		nfcvd 2179	. . . . . . . 8 |- ( F/_ x A -> F/_ x y )
12		id 19	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
13	11, 12	nfeqd 2192	. . . . . . 7 |- ( F/_ x A -> F/ x y = A )
14	13	nfrd 1413	. . . . . 6 |- ( F/_ x A -> ( y = A -> A. x y = A ) )
15	14	eximdv 1760	. . . . 5 |- ( F/_ x A -> ( E. y y = A -> E. y A. x y = A ) )
16	10, 15	syl5bi 141	. . . 4 |- ( F/_ x A -> ( A e. _V -> E. y A. x y = A ) )
17	16	imp 115	. . 3 |- ( ( F/_ x A /\ A e. _V ) -> E. y A. x y = A )
18		eusv1 4184	. . 3 |- ( E! y A. x y = A <-> E. y A. x y = A )
19	17, 18	sylibr 137	. 2 |- ( ( F/_ x A /\ A e. _V ) -> E! y A. x y = A )
20	9, 19	impbii 117	1 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )

Syntax hints:   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   F/_wnfc 2165   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  eusv2nf  4188  eusv2  4189