Theorem eusvobj2 6289

Description: Specify the same property in two ways when class () is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
eusvobj1.1

Assertion

Ref	Expression
eusvobj2

Distinct variable groups:   ,,   ,

Proof of Theorem eusvobj2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 4101	. . 3
2		eleq2 2530	. . . . . 6
3		abid 2444	. . . . . 6
4		elsn 4043	. . . . . 6
5	2, 3, 4	3bitr3g 287	. . . . 5
6		nfre1 2918	. . . . . . . . 9
7	6	nfab 2623	. . . . . . . 8
8	7	nfeq1 2634	. . . . . . 7
9		eusvobj1.1	. . . . . . . . 9
10	9	elabrex 6155	. . . . . . . 8
11		eleq2 2530	. . . . . . . . 9
12	9	elsnc 4053	. . . . . . . . . 10
13		eqcom 2466	. . . . . . . . . 10
14	12, 13	bitri 249	. . . . . . . . 9
15	11, 14	syl6bb 261	. . . . . . . 8
16	10, 15	syl5ib 219	. . . . . . 7
17	8, 16	ralrimi 2857	. . . . . 6
18		eqeq1 2461	. . . . . . 7
19	18	ralbidv 2896	. . . . . 6
20	17, 19	syl5ibrcom 222	. . . . 5
21	5, 20	sylbid 215	. . . 4
22	21	exlimiv 1722	. . 3
23	1, 22	sylbi 195	. 2
24		euex 2308	. . 3
25		rexn0 3932	. . . 4
26	25	exlimiv 1722	. . 3
27		r19.2z 3918	. . . 4
28	27	ex 434	. . 3
29	24, 26, 28	3syl 20	. 2
30	23, 29	impbid 191	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282  {cab 2442  =/=wne 2652  A.wral 2807  E.wrex 2808  cvv 3109  c0 3784  {csn 4029

This theorem is referenced by:  eusvobj1  6290

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-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-nul 3785  df-sn 4030