Step |
Hyp |
Ref |
Expression |
1 |
|
df-3or |
|- ( ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) <-> ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) \/ ( ps /\ ch ) ) ) |
2 |
|
cador |
|- ( cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) ) |
3 |
|
andi |
|- ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) ) |
4 |
3
|
orbi1i |
|- ( ( ( ph /\ ( ps \/ ch ) ) \/ ( ps /\ ch ) ) <-> ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) \/ ( ps /\ ch ) ) ) |
5 |
1 2 4
|
3bitr4i |
|- ( cadd ( ph , ps , ch ) <-> ( ( ph /\ ( ps \/ ch ) ) \/ ( ps /\ ch ) ) ) |
6 |
|
ordir |
|- ( ( ( ph /\ ( ps \/ ch ) ) \/ ( ps /\ ch ) ) <-> ( ( ph \/ ( ps /\ ch ) ) /\ ( ( ps \/ ch ) \/ ( ps /\ ch ) ) ) ) |
7 |
|
ordi |
|- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) ) |
8 |
|
orcom |
|- ( ( ( ps \/ ch ) \/ ( ps /\ ch ) ) <-> ( ( ps /\ ch ) \/ ( ps \/ ch ) ) ) |
9 |
|
animorl |
|- ( ( ps /\ ch ) -> ( ps \/ ch ) ) |
10 |
|
pm4.72 |
|- ( ( ( ps /\ ch ) -> ( ps \/ ch ) ) <-> ( ( ps \/ ch ) <-> ( ( ps /\ ch ) \/ ( ps \/ ch ) ) ) ) |
11 |
9 10
|
mpbi |
|- ( ( ps \/ ch ) <-> ( ( ps /\ ch ) \/ ( ps \/ ch ) ) ) |
12 |
8 11
|
bitr4i |
|- ( ( ( ps \/ ch ) \/ ( ps /\ ch ) ) <-> ( ps \/ ch ) ) |
13 |
7 12
|
anbi12i |
|- ( ( ( ph \/ ( ps /\ ch ) ) /\ ( ( ps \/ ch ) \/ ( ps /\ ch ) ) ) <-> ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ps \/ ch ) ) ) |
14 |
5 6 13
|
3bitri |
|- ( cadd ( ph , ps , ch ) <-> ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ps \/ ch ) ) ) |
15 |
|
df-3an |
|- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ( ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ps \/ ch ) ) ) |
16 |
14 15
|
bitr4i |
|- ( cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) ) |