Step |
Hyp |
Ref |
Expression |
1 |
|
df-nan |
|- ( ( th -/\ ch ) <-> -. ( th /\ ch ) ) |
2 |
|
pm4.57 |
|- ( -. ( -. ( ch /\ th ) /\ -. ( ph /\ th ) ) <-> ( ( ch /\ th ) \/ ( ph /\ th ) ) ) |
3 |
|
orel2 |
|- ( -. ph -> ( ( ch \/ ph ) -> ch ) ) |
4 |
3
|
com12 |
|- ( ( ch \/ ph ) -> ( -. ph -> ch ) ) |
5 |
|
simpr |
|- ( ( ps /\ ch ) -> ch ) |
6 |
5
|
a1i |
|- ( ( ch \/ ph ) -> ( ( ps /\ ch ) -> ch ) ) |
7 |
4 6
|
jad |
|- ( ( ch \/ ph ) -> ( ( ph -> ( ps /\ ch ) ) -> ch ) ) |
8 |
7
|
com12 |
|- ( ( ph -> ( ps /\ ch ) ) -> ( ( ch \/ ph ) -> ch ) ) |
9 |
|
pm3.45 |
|- ( ( ch -> ch ) -> ( ( ch /\ th ) -> ( ch /\ th ) ) ) |
10 |
|
pm3.45 |
|- ( ( ph -> ch ) -> ( ( ph /\ th ) -> ( ch /\ th ) ) ) |
11 |
9 10
|
anim12i |
|- ( ( ( ch -> ch ) /\ ( ph -> ch ) ) -> ( ( ( ch /\ th ) -> ( ch /\ th ) ) /\ ( ( ph /\ th ) -> ( ch /\ th ) ) ) ) |
12 |
|
jaob |
|- ( ( ( ch \/ ph ) -> ch ) <-> ( ( ch -> ch ) /\ ( ph -> ch ) ) ) |
13 |
|
jaob |
|- ( ( ( ( ch /\ th ) \/ ( ph /\ th ) ) -> ( ch /\ th ) ) <-> ( ( ( ch /\ th ) -> ( ch /\ th ) ) /\ ( ( ph /\ th ) -> ( ch /\ th ) ) ) ) |
14 |
11 12 13
|
3imtr4i |
|- ( ( ( ch \/ ph ) -> ch ) -> ( ( ( ch /\ th ) \/ ( ph /\ th ) ) -> ( ch /\ th ) ) ) |
15 |
8 14
|
syl |
|- ( ( ph -> ( ps /\ ch ) ) -> ( ( ( ch /\ th ) \/ ( ph /\ th ) ) -> ( ch /\ th ) ) ) |
16 |
|
pm3.22 |
|- ( ( ch /\ th ) -> ( th /\ ch ) ) |
17 |
15 16
|
syl6 |
|- ( ( ph -> ( ps /\ ch ) ) -> ( ( ( ch /\ th ) \/ ( ph /\ th ) ) -> ( th /\ ch ) ) ) |
18 |
2 17
|
syl5bi |
|- ( ( ph -> ( ps /\ ch ) ) -> ( -. ( -. ( ch /\ th ) /\ -. ( ph /\ th ) ) -> ( th /\ ch ) ) ) |
19 |
18
|
con1d |
|- ( ( ph -> ( ps /\ ch ) ) -> ( -. ( th /\ ch ) -> ( -. ( ch /\ th ) /\ -. ( ph /\ th ) ) ) ) |
20 |
|
df-nan |
|- ( ( ch -/\ th ) <-> -. ( ch /\ th ) ) |
21 |
20
|
biimpri |
|- ( -. ( ch /\ th ) -> ( ch -/\ th ) ) |
22 |
|
df-nan |
|- ( ( ph -/\ th ) <-> -. ( ph /\ th ) ) |
23 |
22
|
biimpri |
|- ( -. ( ph /\ th ) -> ( ph -/\ th ) ) |
24 |
21 23
|
anim12i |
|- ( ( -. ( ch /\ th ) /\ -. ( ph /\ th ) ) -> ( ( ch -/\ th ) /\ ( ph -/\ th ) ) ) |
25 |
19 24
|
syl6 |
|- ( ( ph -> ( ps /\ ch ) ) -> ( -. ( th /\ ch ) -> ( ( ch -/\ th ) /\ ( ph -/\ th ) ) ) ) |
26 |
1 25
|
syl5bi |
|- ( ( ph -> ( ps /\ ch ) ) -> ( ( th -/\ ch ) -> ( ( ch -/\ th ) /\ ( ph -/\ th ) ) ) ) |
27 |
|
nannan |
|- ( ( ph -/\ ( ps -/\ ch ) ) <-> ( ph -> ( ps /\ ch ) ) ) |
28 |
|
nannan |
|- ( ( ( th -/\ ch ) -/\ ( ( ch -/\ th ) -/\ ( ph -/\ th ) ) ) <-> ( ( th -/\ ch ) -> ( ( ch -/\ th ) /\ ( ph -/\ th ) ) ) ) |
29 |
26 27 28
|
3imtr4i |
|- ( ( ph -/\ ( ps -/\ ch ) ) -> ( ( th -/\ ch ) -/\ ( ( ch -/\ th ) -/\ ( ph -/\ th ) ) ) ) |
30 |
29
|
ancli |
|- ( ( ph -/\ ( ps -/\ ch ) ) -> ( ( ph -/\ ( ps -/\ ch ) ) /\ ( ( th -/\ ch ) -/\ ( ( ch -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) |
31 |
|
nannan |
|- ( ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ch -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) <-> ( ( ph -/\ ( ps -/\ ch ) ) -> ( ( ph -/\ ( ps -/\ ch ) ) /\ ( ( th -/\ ch ) -/\ ( ( ch -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) ) |
32 |
30 31
|
mpbir |
|- ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ch -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) |