Step |
Hyp |
Ref |
Expression |
1 |
|
wfisg.1 |
|- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) ) |
2 |
|
wefr |
|- ( R We A -> R Fr A ) |
3 |
2
|
adantr |
|- ( ( R We A /\ R Se A ) -> R Fr A ) |
4 |
|
weso |
|- ( R We A -> R Or A ) |
5 |
|
sopo |
|- ( R Or A -> R Po A ) |
6 |
4 5
|
syl |
|- ( R We A -> R Po A ) |
7 |
6
|
adantr |
|- ( ( R We A /\ R Se A ) -> R Po A ) |
8 |
|
simpr |
|- ( ( R We A /\ R Se A ) -> R Se A ) |
9 |
1
|
adantl |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ y e. A ) -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) ) |
10 |
9
|
frpoinsg |
|- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph ) |
11 |
3 7 8 10
|
syl3anc |
|- ( ( R We A /\ R Se A ) -> A. y e. A ph ) |