Theorem dmdi 27419

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 27416	. . . . 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 3511	. . . . . 6 |- ( x = C -> ( B C_ x <-> B C_ C ) )
4		ineq1 3679	. . . . . . . 8 |- ( x = C -> ( x i^i A ) = ( C i^i A ) )
5	4	oveq1d 6285	. . . . . . 7 |- ( x = C -> ( ( x i^i A ) vH B ) = ( ( C i^i A ) vH B ) )
6		ineq1 3679	. . . . . . 7 |- ( x = C -> ( x i^i ( A vH B ) ) = ( C i^i ( A vH B ) ) )
7	5, 6	eqeq12d 2476	. . . . . 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 318	. . . . 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 3203	. . . 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 655	. . 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 1189	. 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 431	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   i^i cin 3460   C_ wss 3461   class class class wbr 4439  (class class class)co 6270   CHcch 26044   vH chj 26048   MH* cdmd 26082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-iota 5534  df-fv 5578  df-ov 6273  df-dmd 27398

This theorem is referenced by:  dmdi2  27421  dmdsl3  27432  csmdsymi  27451  mdsymlem1  27520