Step |
Hyp |
Ref |
Expression |
1 |
|
sp |
|- ( A. x x = y -> x = y ) |
2 |
|
sbequ2 |
|- ( x = y -> ( [ y / x ] ph -> ph ) ) |
3 |
2
|
sps |
|- ( A. x x = y -> ( [ y / x ] ph -> ph ) ) |
4 |
|
orc |
|- ( ( x = y /\ ph ) -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) |
5 |
1 3 4
|
syl6an |
|- ( A. x x = y -> ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) ) |
6 |
|
sb4b |
|- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) |
7 |
|
olc |
|- ( A. x ( x = y -> ph ) -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) |
8 |
6 7
|
syl6bi |
|- ( -. A. x x = y -> ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) ) |
9 |
5 8
|
pm2.61i |
|- ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) |
10 |
|
sbequ1 |
|- ( x = y -> ( ph -> [ y / x ] ph ) ) |
11 |
10
|
imp |
|- ( ( x = y /\ ph ) -> [ y / x ] ph ) |
12 |
|
sb2 |
|- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) |
13 |
11 12
|
jaoi |
|- ( ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) -> [ y / x ] ph ) |
14 |
9 13
|
impbii |
|- ( [ y / x ] ph <-> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) |