Theorem eusvnf 4290

Description: Even if x is free in A, it is effectively bound when A ( x ) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
eusvnf	|- ( E! y A. x y = A -> F/_ x A )

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem eusvnf

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		euex 1979	. 2 |- ( E! y A. x y = A -> E. y A. x y = A )
2		vex 2625	. . . . . . 7 |- z e. _V
3		nfcv 2229	. . . . . . . 8 |- F/_ x z
4		nfcsb1v 2966	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
5	4	nfeq2 2241	. . . . . . . 8 |- F/ x y = [_ z / x ]_ A
6		csbeq1a 2944	. . . . . . . . 9 |- ( x = z -> A = [_ z / x ]_ A )
7	6	eqeq2d 2100	. . . . . . . 8 |- ( x = z -> ( y = A <-> y = [_ z / x ]_ A ) )
8	3, 5, 7	spcgf 2704	. . . . . . 7 |- ( z e. _V -> ( A. x y = A -> y = [_ z / x ]_ A ) )
9	2, 8	ax-mp 7	. . . . . 6 |- ( A. x y = A -> y = [_ z / x ]_ A )
10		vex 2625	. . . . . . 7 |- w e. _V
11		nfcv 2229	. . . . . . . 8 |- F/_ x w
12		nfcsb1v 2966	. . . . . . . . 9 |- F/_ x [_ w / x ]_ A
13	12	nfeq2 2241	. . . . . . . 8 |- F/ x y = [_ w / x ]_ A
14		csbeq1a 2944	. . . . . . . . 9 |- ( x = w -> A = [_ w / x ]_ A )
15	14	eqeq2d 2100	. . . . . . . 8 |- ( x = w -> ( y = A <-> y = [_ w / x ]_ A ) )
16	11, 13, 15	spcgf 2704	. . . . . . 7 |- ( w e. _V -> ( A. x y = A -> y = [_ w / x ]_ A ) )
17	10, 16	ax-mp 7	. . . . . 6 |- ( A. x y = A -> y = [_ w / x ]_ A )
18	9, 17	eqtr3d 2123	. . . . 5 |- ( A. x y = A -> [_ z / x ]_ A = [_ w / x ]_ A )
19	18	alrimivv 1804	. . . 4 |- ( A. x y = A -> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
20		sbnfc2 2991	. . . 4 |- ( F/_ x A <-> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
21	19, 20	sylibr 133	. . 3 |- ( A. x y = A -> F/_ x A )
22	21	exlimiv 1535	. 2 |- ( E. y A. x y = A -> F/_ x A )
23	1, 22	syl 14	1 |- ( E! y A. x y = A -> F/_ x A )

Syntax hints:   -> wi 4   A.wal 1288   = wceq 1290   E.wex 1427   e. wcel 1439   E!weu 1949   F/_wnfc 2216   _Vcvv 2622   [_csb 2936

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-sbc 2844  df-csb 2937

This theorem is referenced by:  eusvnfb  4291  eusv2i  4292