Theorem mdbr3 29156

Description: Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

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 groups:   x, A   x, B

Proof of Theorem mdbr3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdbr 29153	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
2		chincl 28358	. . . . . . . 8 |- ( ( x e. CH /\ B e. CH ) -> ( x i^i B ) e. CH )
3		inss2 3834	. . . . . . . . 9 |- ( x i^i B ) C_ B
4		sseq1 3626	. . . . . . . . . . 11 |- ( y = ( x i^i B ) -> ( y C_ B <-> ( x i^i B ) C_ B ) )
5		oveq1 6657	. . . . . . . . . . . . 13 |- ( y = ( x i^i B ) -> ( y vH A ) = ( ( x i^i B ) vH A ) )
6	5	ineq1d 3813	. . . . . . . . . . . 12 |- ( y = ( x i^i B ) -> ( ( y vH A ) i^i B ) = ( ( ( x i^i B ) vH A ) i^i B ) )
7		oveq1 6657	. . . . . . . . . . . 12 |- ( y = ( x i^i B ) -> ( y vH ( A i^i B ) ) = ( ( x i^i B ) vH ( A i^i B ) ) )
8	6, 7	eqeq12d 2637	. . . . . . . . . . 11 |- ( 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 ) ) ) )
9	4, 8	imbi12d 334	. . . . . . . . . 10 |- ( y = ( x i^i B ) -> ( ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) <-> ( ( x i^i B ) C_ B -> ( ( ( x i^i B ) vH A ) i^i B ) = ( ( x i^i B ) vH ( A i^i B ) ) ) ) )
10	9	rspcv 3305	. . . . . . . . 9 |- ( ( x i^i B ) e. CH -> ( A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) -> ( ( x i^i B ) C_ B -> ( ( ( x i^i B ) vH A ) i^i B ) = ( ( x i^i B ) vH ( A i^i B ) ) ) ) )
11	3, 10	mpii 46	. . . . . . . 8 |- ( ( x i^i B ) e. CH -> ( A. y e. CH ( y C_ 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 ) ) ) )
12	2, 11	syl 17	. . . . . . 7 |- ( ( x e. CH /\ B e. CH ) -> ( A. y e. CH ( y C_ 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 ) ) ) )
13	12	ex 450	. . . . . 6 |- ( x e. CH -> ( B e. CH -> ( A. y e. CH ( y C_ 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 ) ) ) ) )
14	13	com3l 89	. . . . 5 |- ( B e. CH -> ( A. y e. CH ( y C_ 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 ) ) ) ) )
15	14	ralrimdv 2968	. . . 4 |- ( B e. CH -> ( A. y e. CH ( y C_ 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 ) ) ) )
16		dfss 3589	. . . . . . . . . . 11 |- ( x C_ B <-> x = ( x i^i B ) )
17	16	biimpi 206	. . . . . . . . . 10 |- ( x C_ B -> x = ( x i^i B ) )
18	17	oveq1d 6665	. . . . . . . . 9 |- ( x C_ B -> ( x vH A ) = ( ( x i^i B ) vH A ) )
19	18	ineq1d 3813	. . . . . . . 8 |- ( x C_ B -> ( ( x vH A ) i^i B ) = ( ( ( x i^i B ) vH A ) i^i B ) )
20	17	oveq1d 6665	. . . . . . . 8 |- ( x C_ B -> ( x vH ( A i^i B ) ) = ( ( x i^i B ) vH ( A i^i B ) ) )
21	19, 20	eqeq12d 2637	. . . . . . 7 |- ( x C_ 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 ) ) ) )
22	21	biimprcd 240	. . . . . 6 |- ( ( ( ( x i^i B ) vH A ) i^i B ) = ( ( x i^i B ) vH ( A i^i B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) )
23	22	ralimi 2952	. . . . 5 |- ( 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 C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) )
24		sseq1 3626	. . . . . . 7 |- ( x = y -> ( x C_ B <-> y C_ B ) )
25		oveq1 6657	. . . . . . . . 9 |- ( x = y -> ( x vH A ) = ( y vH A ) )
26	25	ineq1d 3813	. . . . . . . 8 |- ( x = y -> ( ( x vH A ) i^i B ) = ( ( y vH A ) i^i B ) )
27		oveq1 6657	. . . . . . . 8 |- ( x = y -> ( x vH ( A i^i B ) ) = ( y vH ( A i^i B ) ) )
28	26, 27	eqeq12d 2637	. . . . . . 7 |- ( 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 ) ) ) )
29	24, 28	imbi12d 334	. . . . . 6 |- ( x = y -> ( ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) <-> ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
30	29	cbvralv 3171	. . . . 5 |- ( A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) <-> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
31	23, 30	sylib 208	. . . 4 |- ( 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 C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
32	15, 31	impbid1 215	. . 3 |- ( B e. CH -> ( A. y e. CH ( y C_ 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 ) ) ) )
33	32	adantl 482	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( A. y e. CH ( y C_ 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 ) ) ) )
34	1, 33	bitrd 268	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 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   CHcch 27786   vH chj 27790   MH cmd 27823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009  ax-hilex 27856  ax-hfvadd 27857  ax-hv0cl 27860  ax-hfvmul 27862

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-nn 11021  df-hlim 27829  df-sh 28064  df-ch 28078  df-md 29139

This theorem is referenced by: (None)