Step |
Hyp |
Ref |
Expression |
1 |
|
biimp |
|- ( ( ph <-> ps ) -> ( ph -> ps ) ) |
2 |
1
|
imim3i |
|- ( ( x = A -> ( ph <-> ps ) ) -> ( ( x = A -> ph ) -> ( x = A -> ps ) ) ) |
3 |
2
|
al2imi |
|- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A. x ( x = A -> ph ) -> A. x ( x = A -> ps ) ) ) |
4 |
|
elisset |
|- ( A e. V -> E. x x = A ) |
5 |
|
19.23t |
|- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) ) |
6 |
5
|
biimpd |
|- ( F/ x ps -> ( A. x ( x = A -> ps ) -> ( E. x x = A -> ps ) ) ) |
7 |
4 6
|
syl7 |
|- ( F/ x ps -> ( A. x ( x = A -> ps ) -> ( A e. V -> ps ) ) ) |
8 |
3 7
|
sylan9r |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A. x ( x = A -> ph ) -> ( A e. V -> ps ) ) ) |
9 |
8
|
com23 |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. V -> ( A. x ( x = A -> ph ) -> ps ) ) ) |
10 |
9
|
3impia |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) -> ps ) ) |
11 |
|
ceqsal1t |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) ) |
12 |
11
|
3adant3 |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( ps -> A. x ( x = A -> ph ) ) ) |
13 |
10 12
|
impbid |
|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) ) |