Theorem dmdbr3 10227

Description: Binary relation expressing the dual modular pair property. This version quantifies an equality instead of an inference.

Assertion

Ref	Expression
dmdbr3	|- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))

Distinct variable groups:   x,A   x,B

Proof of Theorem dmdbr3

Step	Hyp	Ref	Expression
1		dmdbrt 10221	. 2 |- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B)))))
2		chub2t 9426	. . . . . . . . 9 |- ((B e. CH /\ x e. CH) -> B (_ (x vH B))
3	2	ancoms 438	. . . . . . . 8 |- ((x e. CH /\ B e. CH) -> B (_ (x vH B))
4		chjclt 9324	. . . . . . . . 9 |- ((x e. CH /\ B e. CH) -> (x vH B) e. CH)
5		sseq2 2086	. . . . . . . . . . 11 |- (y = (x vH B) -> (B (_ y <-> B (_ (x vH B)))
6		ineq1 2213	. . . . . . . . . . . . 13 |- (y = (x vH B) -> (y i^i A) = ((x vH B) i^i A))
7	6	opreq1d 3981	. . . . . . . . . . . 12 |- (y = (x vH B) -> ((y i^i A) vH B) = (((x vH B) i^i A) vH B))
8		ineq1 2213	. . . . . . . . . . . 12 |- (y = (x vH B) -> (y i^i (A vH B)) = ((x vH B) i^i (A vH B)))
9	7, 8	eqeq12d 1492	. . . . . . . . . . 11 |- (y = (x vH B) -> (((y i^i A) vH B) = (y i^i (A vH B)) <-> (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))
10	5, 9	imbi12d 628	. . . . . . . . . 10 |- (y = (x vH B) -> ((B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) <-> (B (_ (x vH B) -> (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)))))
11	10	rcla4v 1876	. . . . . . . . 9 |- ((x vH B) e. CH -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) -> (B (_ (x vH B) -> (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)))))
12	4, 11	syl 10	. . . . . . . 8 |- ((x e. CH /\ B e. CH) -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) -> (B (_ (x vH B) -> (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)))))
13	3, 12	mpid 47	. . . . . . 7 |- ((x e. CH /\ B e. CH) -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) -> (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))
14	13	ex 373	. . . . . 6 |- (x e. CH -> (B e. CH -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) -> (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)))))
15	14	com3l 34	. . . . 5 |- (B e. CH -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) -> (x e. CH -> (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)))))
16	15	r19.21adv 1721	. . . 4 |- (B e. CH -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) -> A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))
17		chlejb2t 9431	. . . . . . . . . . . . 13 |- ((B e. CH /\ x e. CH) -> (B (_ x <-> (x vH B) = x))
18	17	biimpa 418	. . . . . . . . . . . 12 |- (((B e. CH /\ x e. CH) /\ B (_ x) -> (x vH B) = x)
19	18	ineq1d 2219	. . . . . . . . . . 11 |- (((B e. CH /\ x e. CH) /\ B (_ x) -> ((x vH B) i^i A) = (x i^i A))
20	19	opreq1d 3981	. . . . . . . . . 10 |- (((B e. CH /\ x e. CH) /\ B (_ x) -> (((x vH B) i^i A) vH B) = ((x i^i A) vH B))
21	18	ineq1d 2219	. . . . . . . . . 10 |- (((B e. CH /\ x e. CH) /\ B (_ x) -> ((x vH B) i^i (A vH B)) = (x i^i (A vH B)))
22	20, 21	eqeq12d 1492	. . . . . . . . 9 |- (((B e. CH /\ x e. CH) /\ B (_ x) -> ((((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)) <-> ((x i^i A) vH B) = (x i^i (A vH B))))
23	22	biimpd 153	. . . . . . . 8 |- (((B e. CH /\ x e. CH) /\ B (_ x) -> ((((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)) -> ((x i^i A) vH B) = (x i^i (A vH B))))
24	23	ex 373	. . . . . . 7 |- ((B e. CH /\ x e. CH) -> (B (_ x -> ((((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)) -> ((x i^i A) vH B) = (x i^i (A vH B)))))
25	24	com23 32	. . . . . 6 |- ((B e. CH /\ x e. CH) -> ((((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)) -> (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))
26	25	r19.20dva 1712	. . . . 5 |- (B e. CH -> (A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)) -> A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))
27		sseq2 2086	. . . . . . 7 |- (x = y -> (B (_ x <-> B (_ y))
28		ineq1 2213	. . . . . . . . 9 |- (x = y -> (x i^i A) = (y i^i A))
29	28	opreq1d 3981	. . . . . . . 8 |- (x = y -> ((x i^i A) vH B) = ((y i^i A) vH B))
30		ineq1 2213	. . . . . . . 8 |- (x = y -> (x i^i (A vH B)) = (y i^i (A vH B)))
31	29, 30	eqeq12d 1492	. . . . . . 7 |- (x = y -> (((x i^i A) vH B) = (x i^i (A vH B)) <-> ((y i^i A) vH B) = (y i^i (A vH B))))
32	27, 31	imbi12d 628	. . . . . 6 |- (x = y -> ((B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B))) <-> (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B)))))
33	32	cbvralv 1803	. . . . 5 |- (A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B))) <-> A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))))
34	26, 33	syl6ib 212	. . . 4 |- (B e. CH -> (A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B)) -> A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B)))))
35	16, 34	impbid 518	. . 3 |- (B e. CH -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) <-> A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))
36	35	adantl 390	. 2 |- ((A e. CH /\ B e. CH) -> (A.y e. CH (B (_ y -> ((y i^i A) vH B) = (y i^i (A vH B))) <-> A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))
37	1, 36	bitrd 530	1 |- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (((x vH B) i^i A) vH B) = ((x vH B) i^i (A vH B))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648   i^i cin 2049   (_ wss 2050   class class class wbr 2624  (class class class)co 3969  CHcch 8793   vH chj 8797   MH* cdmd 8831

This theorem is referenced by:  dmdbr6at 10345

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947  ax-hcompl 9066

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp