Theorem mdi 5727

Description: Consequence of the modular pair property.

Assertion

Ref	Expression
mdi	|- ((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 mdi

Step	Hyp	Ref	Expression
1		mdbr 5726	. . . . 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		sseq1 1521	. . . . . . 7 |- (x = C -> (x (_ B <-> C (_ B))
3		opreq1 3006	. . . . . . . . 9 |- (x = C -> (x vH A) = (C vH A))
4	3	ineq1d 1644	. . . . . . . 8 |- (x = C -> ((x vH A) i^i B) = ((C vH A) i^i B))
5		opreq1 3006	. . . . . . . 8 |- (x = C -> (x vH (A i^i B)) = (C vH (A i^i B)))
6	4, 5	cleq12d 1115	. . . . . . 7 |- (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))))
7	2, 6	imbi12d 474	. . . . . 6 |- (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)))))
8	7	rcla4v 1402	. . . . 5 |- (A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))) -> (C e. CH -> (C (_ B -> ((C vH A) i^i B) = (C vH (A i^i B)))))
9	1, 8	syl6bi 187	. . . 4 |- ((A e. CH /\ B e. CH) -> (A MH B -> (C e. CH -> (C (_ B -> ((C vH A) i^i B) = (C vH (A i^i B))))))
10	9	com23 32	. . 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	imp 277	. 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	3impa 609	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 2   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092  A.wral 1201   i^i cin 1486   (_ wss 1487   class class class wbr 2054  (class class class)co 3001  CHcch 4968   vH chj 4972   MH cmd 4982

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-md 5713

metamath.org