Theorem dmdi 26925

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 26922	. . . . 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 207	. . . 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 3526	. . . . . 6 |- ( x = C -> ( B C_ x <-> B C_ C ) )
4		ineq1 3693	. . . . . . . 8 |- ( x = C -> ( x i^i A ) = ( C i^i A ) )
5	4	oveq1d 6299	. . . . . . 7 |- ( x = C -> ( ( x i^i A ) vH B ) = ( ( C i^i A ) vH B ) )
6		ineq1 3693	. . . . . . 7 |- ( x = C -> ( x i^i ( A vH B ) ) = ( C i^i ( A vH B ) ) )
7	5, 6	eqeq12d 2489	. . . . . 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 320	. . . . 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 3210	. . . 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 657	. . 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 1191	. 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 433	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 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   i^i cin 3475   C_ wss 3476   class class class wbr 4447  (class class class)co 6284   CHcch 25550   vH chj 25554   MH* cdmd 25588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-iota 5551  df-fv 5596  df-ov 6287  df-dmd 26904

This theorem is referenced by:  dmdi2  26927  dmdsl3  26938  csmdsymi  26957  mdsymlem1  27026