Theorem mdi 29154

Description: Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Assertion

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

Proof of Theorem mdi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdbr 29153	. . . . 5 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
2	1	biimpd 219	. . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B -> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
3		sseq1 3626	. . . . . 6 |- ( x = C -> ( x C_ B <-> C C_ B ) )
4		oveq1 6657	. . . . . . . 8 |- ( x = C -> ( x vH A ) = ( C vH A ) )
5	4	ineq1d 3813	. . . . . . 7 |- ( x = C -> ( ( x vH A ) i^i B ) = ( ( C vH A ) i^i B ) )
6		oveq1 6657	. . . . . . 7 |- ( x = C -> ( x vH ( A i^i B ) ) = ( C vH ( A i^i B ) ) )
7	5, 6	eqeq12d 2637	. . . . . 6 |- ( x = C -> ( ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) <-> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) )
8	3, 7	imbi12d 334	. . . . 5 |- ( x = C -> ( ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) <-> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
9	8	rspcv 3305	. . . 4 |- ( C e. CH -> ( A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) -> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
10	2, 9	sylan9 689	. . 3 |- ( ( ( A e. CH /\ B e. CH ) /\ C e. CH ) -> ( A MH B -> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
11	10	3impa 1259	. 2 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A MH B -> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
12	11	imp32 449	1 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ C C_ B ) ) -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   CHcch 27786   vH chj 27790   MH cmd 27823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-md 29139

This theorem is referenced by:  mdsl3  29175  mdslmd3i  29191  mdexchi  29194  atabsi  29260