Step |
Hyp |
Ref |
Expression |
1 |
|
legval.p |
|- P = ( Base ` G ) |
2 |
|
legval.d |
|- .- = ( dist ` G ) |
3 |
|
legval.i |
|- I = ( Itv ` G ) |
4 |
|
legval.l |
|- .<_ = ( leG ` G ) |
5 |
|
legval.g |
|- ( ph -> G e. TarskiG ) |
6 |
|
legid.a |
|- ( ph -> A e. P ) |
7 |
|
legid.b |
|- ( ph -> B e. P ) |
8 |
1 2 3 5 6 7
|
tgbtwntriv2 |
|- ( ph -> B e. ( A I B ) ) |
9 |
|
eqidd |
|- ( ph -> ( A .- B ) = ( A .- B ) ) |
10 |
|
eleq1 |
|- ( x = B -> ( x e. ( A I B ) <-> B e. ( A I B ) ) ) |
11 |
|
oveq2 |
|- ( x = B -> ( A .- x ) = ( A .- B ) ) |
12 |
11
|
eqeq2d |
|- ( x = B -> ( ( A .- B ) = ( A .- x ) <-> ( A .- B ) = ( A .- B ) ) ) |
13 |
10 12
|
anbi12d |
|- ( x = B -> ( ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) <-> ( B e. ( A I B ) /\ ( A .- B ) = ( A .- B ) ) ) ) |
14 |
13
|
rspcev |
|- ( ( B e. P /\ ( B e. ( A I B ) /\ ( A .- B ) = ( A .- B ) ) ) -> E. x e. P ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) ) |
15 |
7 8 9 14
|
syl12anc |
|- ( ph -> E. x e. P ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) ) |
16 |
1 2 3 4 5 6 7 6 7
|
legov |
|- ( ph -> ( ( A .- B ) .<_ ( A .- B ) <-> E. x e. P ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) ) ) |
17 |
15 16
|
mpbird |
|- ( ph -> ( A .- B ) .<_ ( A .- B ) ) |