Theorem preq12b 4206

Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Hypotheses

Ref	Expression
preq12b.1
preq12b.2
preq12b.3
preq12b.4

Assertion

Ref	Expression
preq12b

Proof of Theorem preq12b

Step	Hyp	Ref	Expression
1		preq12b.1	. . . . . 6
2	1	prid1 4138	. . . . 5
3		eleq2 2530	. . . . 5
4	2, 3	mpbii 211	. . . 4
5	1	elpr 4047	. . . 4
6	4, 5	sylib 196	. . 3
7		preq1 4109	. . . . . . . 8
8	7	eqeq1d 2459	. . . . . . 7
9		preq12b.2	. . . . . . . 8
10		preq12b.4	. . . . . . . 8
11	9, 10	preqr2 4205	. . . . . . 7
12	8, 11	syl6bi 228	. . . . . 6
13	12	com12 31	. . . . 5
14	13	ancld 553	. . . 4
15		prcom 4108	. . . . . . 7
16	15	eqeq2i 2475	. . . . . 6
17		preq1 4109	. . . . . . . . 9
18	17	eqeq1d 2459	. . . . . . . 8
19		preq12b.3	. . . . . . . . 9
20	9, 19	preqr2 4205	. . . . . . . 8
21	18, 20	syl6bi 228	. . . . . . 7
22	21	com12 31	. . . . . 6
23	16, 22	sylbi 195	. . . . 5
24	23	ancld 553	. . . 4
25	14, 24	orim12d 838	. . 3
26	6, 25	mpd 15	. 2
27		preq12 4111	. . 3
28		prcom 4108	. . . . 5
29	17, 28	syl6eq 2514	. . . 4
30		preq1 4109	. . . 4
31	29, 30	sylan9eq 2518	. . 3
32	27, 31	jaoi 379	. 2
33	26, 32	impbii 188	1

Syntax hints:  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109  {cpr 4031

This theorem is referenced by:  prel12  4207  opthpr  4208  preq12bg  4209  preqsn  4213  opeqpr  4749  preleq  8055  axlowdimlem13  24257  wlkdvspthlem  24609  altopthsn  29611

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-sn 4030  df-pr 4032