Theorem eusvnfb 4598

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 4597	. . 3 |- ( E! y A. x y = A -> F/_ x A )
2		euex 2116	. . . 4 |- ( E! y A. x y = A -> E. y A. x y = A )
3		id 19	. . . . . . 7 |- ( y = A -> y = A )
4		vex 2824	. . . . . . 7 |- y e. _V
5	3, 4	eqeltrrdi 2330	. . . . . 6 |- ( y = A -> A e. _V )
6	5	sps 1590	. . . . 5 |- ( A. x y = A -> A e. _V )
7	6	exlimiv 1651	. . . 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 306	. 2 |- ( E! y A. x y = A -> ( F/_ x A /\ A e. _V ) )
10		isset 2828	. . . . 5 |- ( A e. _V <-> E. y y = A )
11		nfcvd 2393	. . . . . . . 8 |- ( F/_ x A -> F/_ x y )
12		id 19	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
13	11, 12	nfeqd 2407	. . . . . . 7 |- ( F/_ x A -> F/ x y = A )
14	13	nfrd 1573	. . . . . 6 |- ( F/_ x A -> ( y = A -> A. x y = A ) )
15	14	eximdv 1933	. . . . 5 |- ( F/_ x A -> ( E. y y = A -> E. y A. x y = A ) )
16	10, 15	biimtrid 152	. . . 4 |- ( F/_ x A -> ( A e. _V -> E. y A. x y = A ) )
17	16	imp 124	. . 3 |- ( ( F/_ x A /\ A e. _V ) -> E. y A. x y = A )
18		eusv1 4596	. . 3 |- ( E! y A. x y = A <-> E. y A. x y = A )
19	17, 18	sylibr 134	. 2 |- ( ( F/_ x A /\ A e. _V ) -> E! y A. x y = A )
20	9, 19	impbii 126	1 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   F/_wnfc 2379   _Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is referenced by:  eusv2nf  4600  eusv2  4601