Theorem dmdi 27967

Description: Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Assertion

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

Proof of Theorem dmdi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmdbr 27964	. . . . 5 |- ( ( A e. CH /\ B e. CH ) -> ( A MH* B <-> A. x e. CH ( B C_ x -> ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) ) ) )
2	1	biimpd 212	. . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A MH* B -> A. x e. CH ( B C_ x -> ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) ) ) )
3		sseq2 3422	. . . . . 6 |- ( x = C -> ( B C_ x <-> B C_ C ) )
4		ineq1 3595	. . . . . . . 8 |- ( x = C -> ( x i^i A ) = ( C i^i A ) )
5	4	oveq1d 6291	. . . . . . 7 |- ( x = C -> ( ( x i^i A ) vH B ) = ( ( C i^i A ) vH B ) )
6		ineq1 3595	. . . . . . 7 |- ( x = C -> ( x i^i ( A vH B ) ) = ( C i^i ( A vH B ) ) )
7	5, 6	eqeq12d 2467	. . . . . 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 326	. . . . 5 |- ( x = C -> ( ( B C_ x -> ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) ) <-> ( B C_ C -> ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) ) ) )
9	8	rspcv 3114	. . . 4 |- ( C e. CH -> ( A. x e. CH ( B C_ x -> ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) ) -> ( B C_ C -> ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) ) ) )
10	2, 9	sylan9 667	. . 3 |- ( ( ( A e. CH /\ B e. CH ) /\ C e. CH ) -> ( A MH* B -> ( B C_ C -> ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) ) ) )
11	10	3impa 1205	. 2 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A MH* B -> ( B C_ C -> ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) ) ) )
12	11	imp32 439	1 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH* B /\ B C_ C ) ) -> ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) )

Syntax hints:   -> wi 4   /\ wa 375   /\ w3a 986   = wceq 1448   e. wcel 1891   A.wral 2737   i^i cin 3371   C_ wss 3372   class class class wbr 4374  (class class class)co 6276   CHcch 26594   vH chj 26598   MH* cdmd 26632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-iota 5525  df-fv 5569  df-ov 6279  df-dmd 27946

This theorem is referenced by:  dmdi2  27969  dmdsl3  27980  csmdsymi  27999  mdsymlem1  28068