Theorem dmdit 10220

Description: Consequence of the dual modular pair property.

Assertion

Ref	Expression
dmdit	|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A MH* B /\ B (_ C)) -> ((C i^i A) vH B) = (C i^i (A vH B)))

Proof of Theorem dmdit

Step	Hyp	Ref	Expression
1		dmdbrt 10217	. . . . 5 |- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))
2	1	biimpd 153	. . . 4 |- ((A e. CH /\ B e. CH) -> (A MH* B -> A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))
3		sseq2 2081	. . . . . 6 |- (x = C -> (B (_ x <-> B (_ C))
4		ineq1 2208	. . . . . . . 8 |- (x = C -> (x i^i A) = (C i^i A))
5	4	opreq1d 3972	. . . . . . 7 |- (x = C -> ((x i^i A) vH B) = ((C i^i A) vH B))
6		ineq1 2208	. . . . . . 7 |- (x = C -> (x i^i (A vH B)) = (C i^i (A vH B)))
7	5, 6	eqeq12d 1488	. . . . . 6 |- (x = C -> (((x i^i A) vH B) = (x i^i (A vH B)) <-> ((C i^i A) vH B) = (C i^i (A vH B))))
8	3, 7	imbi12d 625	. . . . 5 |- (x = C -> ((B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B))) <-> (B (_ C -> ((C i^i A) vH B) = (C i^i (A vH B)))))
9	8	rcla4v 1871	. . . 4 |- (C e. CH -> (A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B))) -> (B (_ C -> ((C i^i A) vH B) = (C i^i (A vH B)))))
10	2, 9	sylan9 468	. . 3 |- (((A e. CH /\ B e. CH) /\ C e. CH) -> (A MH* B -> (B (_ C -> ((C i^i A) vH B) = (C i^i (A vH B)))))
11	10	3impa 827	. 2 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A MH* B -> (B (_ C -> ((C i^i A) vH B) = (C i^i (A vH B)))))
12	11	imp32 363	1 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A MH* B /\ B (_ C)) -> ((C i^i A) vH B) = (C i^i (A vH B)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  A.wral 1644   i^i cin 2044   (_ wss 2045   class class class wbr 2616  (class class class)co 3960  CHcch 8782   vH chj 8786   MH* cdmd 8820

This theorem is referenced by:  dmdi2 10222  dmdsl3t 10233  csmdsym 10252  mdsymlem1 10321

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-xp 3181  df-cnv 3183  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fv 3195  df-opr 3962  df-dmd 10199

Copyright terms: Public domain