Step |
Hyp |
Ref |
Expression |
1 |
|
elissetv |
|- ( A e. V -> E. y y = A ) |
2 |
|
cbvexeqsetf |
|- ( F/_ x A -> ( E. x x = A <-> E. y y = A ) ) |
3 |
1 2
|
imbitrrid |
|- ( F/_ x A -> ( A e. V -> E. x x = A ) ) |
4 |
|
pm2.04 |
|- ( ( x = A -> ( ph -> ps ) ) -> ( ph -> ( x = A -> ps ) ) ) |
5 |
4
|
al2imi |
|- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A. x ph -> A. x ( x = A -> ps ) ) ) |
6 |
|
19.23t |
|- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) ) |
7 |
6
|
biimpd |
|- ( F/ x ps -> ( A. x ( x = A -> ps ) -> ( E. x x = A -> ps ) ) ) |
8 |
5 7
|
sylan9r |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph -> ps ) ) ) -> ( A. x ph -> ( E. x x = A -> ps ) ) ) |
9 |
8
|
com23 |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph -> ps ) ) ) -> ( E. x x = A -> ( A. x ph -> ps ) ) ) |
10 |
3 9
|
sylan9 |
|- ( ( F/_ x A /\ ( F/ x ps /\ A. x ( x = A -> ( ph -> ps ) ) ) ) -> ( A e. V -> ( A. x ph -> ps ) ) ) |
11 |
10
|
anassrs |
|- ( ( ( F/_ x A /\ F/ x ps ) /\ A. x ( x = A -> ( ph -> ps ) ) ) -> ( A e. V -> ( A. x ph -> ps ) ) ) |