Theorem mdbr3 5729

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

Assertion

Ref	Expression
mdbr3	|- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))

Distinct variable group(s):   x,A   x,B

Proof of Theorem mdbr3

Step	Hyp	Ref	Expression
1		mdbr 5726	. 2 |- ((A e. CH /\ B e. CH) -> (A MH B <-> A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B)))))
2		chinclt 5416	. . . . . . . . . . 11 |- ((x e. CH /\ B e. CH) -> (x i^i B) e. CH)
3	2	ancoms 334	. . . . . . . . . 10 |- ((B e. CH /\ x e. CH) -> (x i^i B) e. CH)
4		inss2 1658	. . . . . . . . . 10 |- (x i^i B) (_ B
5	3, 4	jctir 241	. . . . . . . . 9 |- ((B e. CH /\ x e. CH) -> ((x i^i B) e. CH /\ (x i^i B) (_ B))
6	5	exp 291	. . . . . . . 8 |- (B e. CH -> (x e. CH -> ((x i^i B) e. CH /\ (x i^i B) (_ B)))
7	6	syl4d 28	. . . . . . 7 |- (B e. CH -> ((((x i^i B) e. CH /\ (x i^i B) (_ B) -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))) -> (x e. CH -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)))))
8		impexp 276	. . . . . . 7 |- ((((x i^i B) e. CH /\ (x i^i B) (_ B) -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))) <-> ((x i^i B) e. CH -> ((x i^i B) (_ B -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)))))
9	7, 8	syl5ibr 182	. . . . . 6 |- (B e. CH -> (((x i^i B) e. CH -> ((x i^i B) (_ B -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)))) -> (x e. CH -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)))))
10		sseq1 1521	. . . . . . . 8 |- (y = (x i^i B) -> (y (_ B <-> (x i^i B) (_ B))
11		opreq1 3006	. . . . . . . . . 10 |- (y = (x i^i B) -> (y vH A) = ((x i^i B) vH A))
12	11	ineq1d 1644	. . . . . . . . 9 |- (y = (x i^i B) -> ((y vH A) i^i B) = (((x i^i B) vH A) i^i B))
13		opreq1 3006	. . . . . . . . 9 |- (y = (x i^i B) -> (y vH (A i^i B)) = ((x i^i B) vH (A i^i B)))
14	12, 13	cleq12d 1115	. . . . . . . 8 |- (y = (x i^i B) -> (((y vH A) i^i B) = (y vH (A i^i B)) <-> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))
15	10, 14	imbi12d 474	. . . . . . 7 |- (y = (x i^i B) -> ((y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))) <-> ((x i^i B) (_ B -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)))))
16	15	rcla4v 1402	. . . . . 6 |- (A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))) -> ((x i^i B) e. CH -> ((x i^i B) (_ B -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)))))
17	9, 16	syl5 22	. . . . 5 |- (B e. CH -> (A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))) -> (x e. CH -> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)))))
18	17	r19.21adv 1262	. . . 4 |- (B e. CH -> (A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))) -> A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))
19		dfss 1493	. . . . . . . . . . . 12 |- (x (_ B <-> x = (x i^i B))
20	19	biimp 133	. . . . . . . . . . 11 |- (x (_ B -> x = (x i^i B))
21	20	opreq1d 3012	. . . . . . . . . 10 |- (x (_ B -> (x vH A) = ((x i^i B) vH A))
22	21	ineq1d 1644	. . . . . . . . 9 |- (x (_ B -> ((x vH A) i^i B) = (((x i^i B) vH A) i^i B))
23	20	opreq1d 3012	. . . . . . . . 9 |- (x (_ B -> (x vH (A i^i B)) = ((x i^i B) vH (A i^i B)))
24	22, 23	cleq12d 1115	. . . . . . . 8 |- (x (_ B -> (((x vH A) i^i B) = (x vH (A i^i B)) <-> (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))
25	24	biimprcd 138	. . . . . . 7 |- ((((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)) -> (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))
26	25	r19.20si 1254	. . . . . 6 |- (A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)) -> A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))
27		sseq1 1521	. . . . . . . 8 |- (x = y -> (x (_ B <-> y (_ B))
28		opreq1 3006	. . . . . . . . . 10 |- (x = y -> (x vH A) = (y vH A))
29	28	ineq1d 1644	. . . . . . . . 9 |- (x = y -> ((x vH A) i^i B) = ((y vH A) i^i B))
30		opreq1 3006	. . . . . . . . 9 |- (x = y -> (x vH (A i^i B)) = (y vH (A i^i B)))
31	29, 30	cleq12d 1115	. . . . . . . 8 |- (x = y -> (((x vH A) i^i B) = (x vH (A i^i B)) <-> ((y vH A) i^i B) = (y vH (A i^i B))))
32	27, 31	imbi12d 474	. . . . . . 7 |- (x = y -> ((x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))) <-> (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B)))))
33	32	cbvralv 1333	. . . . . 6 |- (A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))) <-> A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))))
34	26, 33	sylib 173	. . . . 5 |- (A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)) -> A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))))
35	34	a1i 7	. . . 4 |- (B e. CH -> (A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B)) -> A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B)))))
36	18, 35	impbid 397	. . 3 |- (B e. CH -> (A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))) <-> A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))
37	36	adantl 305	. 2 |- ((A e. CH /\ B e. CH) -> (A.y e. CH (y (_ B -> ((y vH A) i^i B) = (y vH (A i^i B))) <-> A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))
38	1, 37	bitrd 406	1 |- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (((x i^i B) vH A) i^i B) = ((x i^i B) vH (A i^i B))))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201   i^i cin 1486   (_ wss 1487   class class class wbr 2054  (class class class)co 3001  CHcch 4968   vH chj 4972   MH cmd 4982

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-hilex 4983  ax-hvaddcl 4984  ax-hvzercl 4987  ax-hvmulcl 4989

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-0r 3965  df-1r 3966  df-c 4034  df-1 4036  df-r 4038  df-plus 4039  df-n 4423  df-hlim 5107  df-sh 5114  df-ch 5127  df-md 5713

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