Theorem eusvnf 4647

Description: Even if is free in , 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

Distinct variable groups:   ,   ,

Proof of Theorem eusvnf

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

Step	Hyp	Ref	Expression
1		euex 2308	. 2
2		vex 3112	. . . . . . 7
3		nfcv 2619	. . . . . . . 8
4		nfcsb1v 3450	. . . . . . . . 9
5	4	nfeq2 2636	. . . . . . . 8
6		csbeq1a 3443	. . . . . . . . 9
7	6	eqeq2d 2471	. . . . . . . 8
8	3, 5, 7	spcgf 3189	. . . . . . 7
9	2, 8	ax-mp 5	. . . . . 6
10		vex 3112	. . . . . . 7
11		nfcv 2619	. . . . . . . 8
12		nfcsb1v 3450	. . . . . . . . 9
13	12	nfeq2 2636	. . . . . . . 8
14		csbeq1a 3443	. . . . . . . . 9
15	14	eqeq2d 2471	. . . . . . . 8
16	11, 13, 15	spcgf 3189	. . . . . . 7
17	10, 16	ax-mp 5	. . . . . 6
18	9, 17	eqtr3d 2500	. . . . 5
19	18	alrimivv 1720	. . . 4
20		sbnfc2 3854	. . . 4
21	19, 20	sylibr 212	. . 3
22	21	exlimiv 1722	. 2
23	1, 22	syl 16	1

Syntax hints:  ->wi 4  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282  F/_wnfc 2605  cvv 3109  [_csb 3434

This theorem is referenced by:  eusvnfb  4648  eusv2i  4649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785