Theorem mdit 10222

Description: Consequence of the modular pair property.

Assertion

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

Proof of Theorem mdit

Step	Hyp	Ref	Expression
1		mdbrt 10221	. . . . 5 |- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
2	1	biimpd 153	. . . 4 |- ((A e. CH /\ B e. CH) -> (A MH B -> A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
3		sseq1 2082	. . . . . 6 |- (x = C -> (x (_ B <-> C (_ B))
4		opreq1 3968	. . . . . . . 8 |- (x = C -> (x vH A) = (C vH A))
5	4	ineq1d 2216	. . . . . . 7 |- (x = C -> ((x vH A) i^i B) = ((C vH A) i^i B))
6		opreq1 3968	. . . . . . 7 |- (x = C -> (x vH (A i^i B)) = (C vH (A i^i B)))
7	5, 6	eqeq12d 1489	. . . . . 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 626	. . . . 5 |- (x = C -> ((x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))) <-> (C (_ B -> ((C vH A) i^i B) = (C vH (A i^i B)))))
9	8	rcla4v 1873	. . . 4 |- (C e. CH -> (A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))) -> (C (_ B -> ((C vH A) i^i B) = (C vH (A i^i B)))))
10	2, 9	sylan9 468	. . 3 |- (((A e. CH /\ B e. CH) /\ C e. CH) -> (A MH B -> (C (_ B -> ((C vH A) i^i B) = (C vH (A i^i B)))))
11	10	3impa 828	. 2 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A MH B -> (C (_ B -> ((C vH A) i^i B) = (C vH (A i^i B)))))
12	11	imp32 363	1 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A MH B /\ C (_ B)) -> ((C vH A) i^i B) = (C vH (A i^i B)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047   class class class wbr 2619  (class class class)co 3963  CHcch 8798   vH chj 8802   MH cmd 8835

This theorem is referenced by:  mdsl3t 10243  mdslmd3 10259  mdexch 10262  atabs 10328

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-md 10207

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