Step |
Hyp |
Ref |
Expression |
1 |
|
df-mo |
|- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) |
2 |
|
sp |
|- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) ) |
3 |
|
pm3.45 |
|- ( ( ph -> x = y ) -> ( ( ph /\ ps ) -> ( x = y /\ ps ) ) ) |
4 |
3
|
aleximi |
|- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> E. x ( x = y /\ ps ) ) ) |
5 |
|
ax12ev2 |
|- ( E. x ( x = y /\ ps ) -> ( x = y -> ps ) ) |
6 |
4 5
|
syl6 |
|- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( x = y -> ps ) ) ) |
7 |
2 6
|
syl5d |
|- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) ) |
8 |
7
|
exlimiv |
|- ( E. y A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) ) |
9 |
1 8
|
sylbi |
|- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) ) |
10 |
9
|
imp |
|- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) |