Theorem eusvnf 4681

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 2281	. 2 |- ( E! y A. x y = A -> E. y A. x y = A )
2		vex 2923	. . . . . . 7 |- z e. _V
3		nfcv 2544	. . . . . . . 8 |- F/_ x z
4		nfcsb1v 3247	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
5	4	nfeq2 2555	. . . . . . . 8 |- F/ x y = [_ z / x ]_ A
6		csbeq1a 3223	. . . . . . . . 9 |- ( x = z -> A = [_ z / x ]_ A )
7	6	eqeq2d 2419	. . . . . . . 8 |- ( x = z -> ( y = A <-> y = [_ z / x ]_ A ) )
8	3, 5, 7	spcgf 2995	. . . . . . 7 |- ( z e. _V -> ( A. x y = A -> y = [_ z / x ]_ A ) )
9	2, 8	ax-mp 8	. . . . . 6 |- ( A. x y = A -> y = [_ z / x ]_ A )
10		vex 2923	. . . . . . 7 |- w e. _V
11		nfcv 2544	. . . . . . . 8 |- F/_ x w
12		nfcsb1v 3247	. . . . . . . . 9 |- F/_ x [_ w / x ]_ A
13	12	nfeq2 2555	. . . . . . . 8 |- F/ x y = [_ w / x ]_ A
14		csbeq1a 3223	. . . . . . . . 9 |- ( x = w -> A = [_ w / x ]_ A )
15	14	eqeq2d 2419	. . . . . . . 8 |- ( x = w -> ( y = A <-> y = [_ w / x ]_ A ) )
16	11, 13, 15	spcgf 2995	. . . . . . 7 |- ( w e. _V -> ( A. x y = A -> y = [_ w / x ]_ A ) )
17	10, 16	ax-mp 8	. . . . . 6 |- ( A. x y = A -> y = [_ w / x ]_ A )
18	9, 17	eqtr3d 2442	. . . . 5 |- ( A. x y = A -> [_ z / x ]_ A = [_ w / x ]_ A )
19	18	alrimivv 1639	. . . 4 |- ( A. x y = A -> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
20		sbnfc2 3273	. . . 4 |- ( F/_ x A <-> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
21	19, 20	sylibr 204	. . 3 |- ( A. x y = A -> F/_ x A )
22	21	exlimiv 1641	. 2 |- ( E. y A. x y = A -> F/_ x A )
23	1, 22	syl 16	1 |- ( E! y A. x y = A -> F/_ x A )

Syntax hints:   -> wi 4   A.wal 1546   E.wex 1547   = wceq 1649   e. wcel 1721   E!weu 2258   F/_wnfc 2531   _Vcvv 2920   [_csb 3215

This theorem is referenced by:  eusvnfb  4682  eusv2i  4683

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-sbc 3126  df-csb 3216