Theorem eusvnf 4431

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 2044	. 2 |- ( E! y A. x y = A -> E. y A. x y = A )
2		vex 2729	. . . . . . 7 |- z e. _V
3		nfcv 2308	. . . . . . . 8 |- F/_ x z
4		nfcsb1v 3078	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
5	4	nfeq2 2320	. . . . . . . 8 |- F/ x y = [_ z / x ]_ A
6		csbeq1a 3054	. . . . . . . . 9 |- ( x = z -> A = [_ z / x ]_ A )
7	6	eqeq2d 2177	. . . . . . . 8 |- ( x = z -> ( y = A <-> y = [_ z / x ]_ A ) )
8	3, 5, 7	spcgf 2808	. . . . . . 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 2729	. . . . . . 7 |- w e. _V
11		nfcv 2308	. . . . . . . 8 |- F/_ x w
12		nfcsb1v 3078	. . . . . . . . 9 |- F/_ x [_ w / x ]_ A
13	12	nfeq2 2320	. . . . . . . 8 |- F/ x y = [_ w / x ]_ A
14		csbeq1a 3054	. . . . . . . . 9 |- ( x = w -> A = [_ w / x ]_ A )
15	14	eqeq2d 2177	. . . . . . . 8 |- ( x = w -> ( y = A <-> y = [_ w / x ]_ A ) )
16	11, 13, 15	spcgf 2808	. . . . . . 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 2200	. . . . 5 |- ( A. x y = A -> [_ z / x ]_ A = [_ w / x ]_ A )
19	18	alrimivv 1863	. . . 4 |- ( A. x y = A -> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
20		sbnfc2 3105	. . . 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 1586	. 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 1341   = wceq 1343   E.wex 1480   E!weu 2014   e. wcel 2136   F/_wnfc 2295   _Vcvv 2726   [_csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  eusvnfb  4432  eusv2i  4433