Theorem wloglei 10110

Description: Form of wlogle 10111 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.)

Hypotheses

Ref	Expression
wlogle.1
wlogle.2
wlogle.3
wloglei.4
wloglei.5

Assertion

Ref	Expression
wloglei

Distinct variable groups:   ,,,,   ,S,,,   ,,   ,,

Proof of Theorem wloglei

Step	Hyp	Ref	Expression
1		wlogle.3	. . . 4
2	1	adantr 465	. . 3
3		simprr 757	. . 3
4	2, 3	sseldd 3504	. 2
5		simprl 756	. . 3
6	2, 5	sseldd 3504	. 2
7		vex 3112	. . 3
8		vex 3112	. . 3
9		eleq1 2529	. . . . . . 7
10		eleq1 2529	. . . . . . 7
11	9, 10	bi2anan9 873	. . . . . 6
12	11	anbi2d 703	. . . . 5
13		breq12 4457	. . . . . 6
14	13	ancoms 453	. . . . 5
15	12, 14	anbi12d 710	. . . 4
16		wlogle.1	. . . 4
17	15, 16	imbi12d 320	. . 3
18		vex 3112	. . . 4
19		vex 3112	. . . 4
20		ancom 450	. . . . . . . 8
21		eleq1 2529	. . . . . . . . 9
22		eleq1 2529	. . . . . . . . 9
23	21, 22	bi2anan9 873	. . . . . . . 8
24	20, 23	syl5bb 257	. . . . . . 7
25	24	anbi2d 703	. . . . . 6
26		breq12 4457	. . . . . . 7
27	26	ancoms 453	. . . . . 6
28	25, 27	anbi12d 710	. . . . 5
29		equcom 1794	. . . . . . 7
30		equcom 1794	. . . . . . 7
31		wlogle.2	. . . . . . 7
32	29, 30, 31	syl2anb 479	. . . . . 6
33	32	bicomd 201	. . . . 5
34	28, 33	imbi12d 320	. . . 4
35		df-3an 975	. . . . . 6
36		wloglei.4	. . . . . 6
37	35, 36	sylan2br 476	. . . . 5
38	37	anassrs 648	. . . 4
39	18, 19, 34, 38	vtocl2 3162	. . 3
40	7, 8, 17, 39	vtocl2 3162	. 2
41		wloglei.5	. . . 4
42	35, 41	sylan2br 476	. . 3
43	42	anassrs 648	. 2
44	4, 6, 40, 43	lecasei 9711	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  e.wcel 1818  C_wss 3475   class class class wbr 4452  cr 9512  cle 9650

This theorem is referenced by:  wlogle  10111  rescon  28691

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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-resscn 9570  ax-pre-lttri 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-er 7330  df-en 7537  df-dom 7538  df-sdom 7539  df-pnf 9651  df-mnf 9652  df-xr 9653  df-ltxr 9654  df-le 9655