Theorem eusvnfb 4862

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 4861	. . 3 |- ( E! y A. x y = A -> F/_ x A )
2		euex 2494	. . . 4 |- ( E! y A. x y = A -> E. y A. x y = A )
3		eqvisset 3211	. . . . . 6 |- ( y = A -> A e. _V )
4	3	sps 2055	. . . . 5 |- ( A. x y = A -> A e. _V )
5	4	exlimiv 1858	. . . 4 |- ( E. y A. x y = A -> A e. _V )
6	2, 5	syl 17	. . 3 |- ( E! y A. x y = A -> A e. _V )
7	1, 6	jca 554	. 2 |- ( E! y A. x y = A -> ( F/_ x A /\ A e. _V ) )
8		isset 3207	. . . . 5 |- ( A e. _V <-> E. y y = A )
9		nfcvd 2765	. . . . . . . 8 |- ( F/_ x A -> F/_ x y )
10		id 22	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
11	9, 10	nfeqd 2772	. . . . . . 7 |- ( F/_ x A -> F/ x y = A )
12	11	nf5rd 2066	. . . . . 6 |- ( F/_ x A -> ( y = A -> A. x y = A ) )
13	12	eximdv 1846	. . . . 5 |- ( F/_ x A -> ( E. y y = A -> E. y A. x y = A ) )
14	8, 13	syl5bi 232	. . . 4 |- ( F/_ x A -> ( A e. _V -> E. y A. x y = A ) )
15	14	imp 445	. . 3 |- ( ( F/_ x A /\ A e. _V ) -> E. y A. x y = A )
16		eusv1 4860	. . 3 |- ( E! y A. x y = A <-> E. y A. x y = A )
17	15, 16	sylibr 224	. 2 |- ( ( F/_ x A /\ A e. _V ) -> E! y A. x y = A )
18	7, 17	impbii 199	1 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  eusv2nf  4864  eusv2  4865