Theorem eusvnf 4185

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 1930	. 2 |- ( E! y A. x y = A -> E. y A. x y = A )
2		vex 2560	. . . . . . 7 |- z e. _V
3		nfcv 2178	. . . . . . . 8 |- F/_ x z
4		nfcsb1v 2882	. . . . . . . . 9 |- F/_ x [_ z / x ]_ A
5	4	nfeq2 2189	. . . . . . . 8 |- F/ x y = [_ z / x ]_ A
6		csbeq1a 2860	. . . . . . . . 9 |- ( x = z -> A = [_ z / x ]_ A )
7	6	eqeq2d 2051	. . . . . . . 8 |- ( x = z -> ( y = A <-> y = [_ z / x ]_ A ) )
8	3, 5, 7	spcgf 2635	. . . . . . 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 2560	. . . . . . 7 |- w e. _V
11		nfcv 2178	. . . . . . . 8 |- F/_ x w
12		nfcsb1v 2882	. . . . . . . . 9 |- F/_ x [_ w / x ]_ A
13	12	nfeq2 2189	. . . . . . . 8 |- F/ x y = [_ w / x ]_ A
14		csbeq1a 2860	. . . . . . . . 9 |- ( x = w -> A = [_ w / x ]_ A )
15	14	eqeq2d 2051	. . . . . . . 8 |- ( x = w -> ( y = A <-> y = [_ w / x ]_ A ) )
16	11, 13, 15	spcgf 2635	. . . . . . 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 2074	. . . . 5 |- ( A. x y = A -> [_ z / x ]_ A = [_ w / x ]_ A )
19	18	alrimivv 1755	. . . 4 |- ( A. x y = A -> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
20		sbnfc2 2906	. . . 4 |- ( F/_ x A <-> A. z A. w [_ z / x ]_ A = [_ w / x ]_ A )
21	19, 20	sylibr 137	. . 3 |- ( A. x y = A -> F/_ x A )
22	21	exlimiv 1489	. 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 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   F/_wnfc 2165   _Vcvv 2557   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  eusvnfb  4186  eusv2i  4187