Theorem dmdi 25711

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 25708	. . . . 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 3383	. . . . . 6 |- ( x = C -> ( B C_ x <-> B C_ C ) )
4		ineq1 3550	. . . . . . . 8 |- ( x = C -> ( x i^i A ) = ( C i^i A ) )
5	4	oveq1d 6111	. . . . . . 7 |- ( x = C -> ( ( x i^i A ) vH B ) = ( ( C i^i A ) vH B ) )
6		ineq1 3550	. . . . . . 7 |- ( x = C -> ( x i^i ( A vH B ) ) = ( C i^i ( A vH B ) ) )
7	5, 6	eqeq12d 2457	. . . . . 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 3074	. . . 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 1182	. 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 965   = wceq 1369   e. wcel 1756   A.wral 2720   i^i cin 3332   C_ wss 3333   class class class wbr 4297  (class class class)co 6096   CHcch 24336   vH chj 24340   MH* cdmd 24374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-iota 5386  df-fv 5431  df-ov 6099  df-dmd 25690

This theorem is referenced by:  dmdi2  25713  dmdsl3  25724  csmdsymi  25743  mdsymlem1  25812