Theorem eusvnf 4550

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 2109	. 2 |- ( E! y A. x y = A -> E. y A. x y = A )
2		vex 2805	. . . . . . 7 |- z e. _V
3		nfcv 2374	. . . . . . . 8 |- F/_ x z
4		nfcsb1v 3160	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
5	4	nfeq2 2386	. . . . . . . 8 |- F/ x y = [_ z / x ]_ A
6		csbeq1a 3136	. . . . . . . . 9 |- ( x = z -> A = [_ z / x ]_ A )
7	6	eqeq2d 2243	. . . . . . . 8 |- ( x = z -> ( y = A <-> y = [_ z / x ]_ A ) )
8	3, 5, 7	spcgf 2888	. . . . . . 7 |- ( z e. _V -> ( A. x y = A -> y = [_ z / x ]_ A ) )
9	2, 8	ax-mp 5	. . . . . 6 |- ( A. x y = A -> y = [_ z / x ]_ A )
10		vex 2805	. . . . . . 7 |- w e. _V
11		nfcv 2374	. . . . . . . 8 |- F/_ x w
12		nfcsb1v 3160	. . . . . . . . 9 |- F/_ x [_ w / x ]_ A
13	12	nfeq2 2386	. . . . . . . 8 |- F/ x y = [_ w / x ]_ A
14		csbeq1a 3136	. . . . . . . . 9 |- ( x = w -> A = [_ w / x ]_ A )
15	14	eqeq2d 2243	. . . . . . . 8 |- ( x = w -> ( y = A <-> y = [_ w / x ]_ A ) )
16	11, 13, 15	spcgf 2888	. . . . . . 7 |- ( w e. _V -> ( A. x y = A -> y = [_ w / x ]_ A ) )
17	10, 16	ax-mp 5	. . . . . 6 |- ( A. x y = A -> y = [_ w / x ]_ A )
18	9, 17	eqtr3d 2266	. . . . 5 |- ( A. x y = A -> [_ z / x ]_ A = [_ w / x ]_ A )
19	18	alrimivv 1923	. . . 4 |- ( A. x y = A -> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
20		sbnfc2 3188	. . . 4 |- ( F/_ x A <-> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
21	19, 20	sylibr 134	. . 3 |- ( A. x y = A -> F/_ x A )
22	21	exlimiv 1646	. 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 1395   = wceq 1397   E.wex 1540   E!weu 2079   e. wcel 2202   F/_wnfc 2361   _Vcvv 2802   [_csb 3127

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032  df-csb 3128

This theorem is referenced by:  eusvnfb  4551  eusv2i  4552