Step |
Hyp |
Ref |
Expression |
1 |
|
sban |
|- ( [ x / b ] ( ph /\ ps ) <-> ( [ x / b ] ph /\ [ x / b ] ps ) ) |
2 |
1
|
sbbii |
|- ( [ b / a ] [ x / b ] ( ph /\ ps ) <-> [ b / a ] ( [ x / b ] ph /\ [ x / b ] ps ) ) |
3 |
2
|
sbbii |
|- ( [ a / x ] [ b / a ] [ x / b ] ( ph /\ ps ) <-> [ a / x ] [ b / a ] ( [ x / b ] ph /\ [ x / b ] ps ) ) |
4 |
|
sban |
|- ( [ b / a ] ( [ x / b ] ph /\ [ x / b ] ps ) <-> ( [ b / a ] [ x / b ] ph /\ [ b / a ] [ x / b ] ps ) ) |
5 |
4
|
sbbii |
|- ( [ a / x ] [ b / a ] ( [ x / b ] ph /\ [ x / b ] ps ) <-> [ a / x ] ( [ b / a ] [ x / b ] ph /\ [ b / a ] [ x / b ] ps ) ) |
6 |
|
sban |
|- ( [ a / x ] ( [ b / a ] [ x / b ] ph /\ [ b / a ] [ x / b ] ps ) <-> ( [ a / x ] [ b / a ] [ x / b ] ph /\ [ a / x ] [ b / a ] [ x / b ] ps ) ) |
7 |
3 5 6
|
3bitri |
|- ( [ a / x ] [ b / a ] [ x / b ] ( ph /\ ps ) <-> ( [ a / x ] [ b / a ] [ x / b ] ph /\ [ a / x ] [ b / a ] [ x / b ] ps ) ) |
8 |
|
pm4.38 |
|- ( ( ( [ a / x ] [ b / a ] [ x / b ] ph <-> ph ) /\ ( [ a / x ] [ b / a ] [ x / b ] ps <-> ps ) ) -> ( ( [ a / x ] [ b / a ] [ x / b ] ph /\ [ a / x ] [ b / a ] [ x / b ] ps ) <-> ( ph /\ ps ) ) ) |
9 |
7 8
|
syl5bb |
|- ( ( ( [ a / x ] [ b / a ] [ x / b ] ph <-> ph ) /\ ( [ a / x ] [ b / a ] [ x / b ] ps <-> ps ) ) -> ( [ a / x ] [ b / a ] [ x / b ] ( ph /\ ps ) <-> ( ph /\ ps ) ) ) |
10 |
9
|
alanimi |
|- ( ( A. b ( [ a / x ] [ b / a ] [ x / b ] ph <-> ph ) /\ A. b ( [ a / x ] [ b / a ] [ x / b ] ps <-> ps ) ) -> A. b ( [ a / x ] [ b / a ] [ x / b ] ( ph /\ ps ) <-> ( ph /\ ps ) ) ) |
11 |
10
|
alanimi |
|- ( ( A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ph <-> ph ) /\ A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ps <-> ps ) ) -> A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ( ph /\ ps ) <-> ( ph /\ ps ) ) ) |
12 |
|
df-ich |
|- ( [ a <> b ] ph <-> A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ph <-> ph ) ) |
13 |
|
df-ich |
|- ( [ a <> b ] ps <-> A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ps <-> ps ) ) |
14 |
12 13
|
anbi12i |
|- ( ( [ a <> b ] ph /\ [ a <> b ] ps ) <-> ( A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ph <-> ph ) /\ A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ps <-> ps ) ) ) |
15 |
|
df-ich |
|- ( [ a <> b ] ( ph /\ ps ) <-> A. a A. b ( [ a / x ] [ b / a ] [ x / b ] ( ph /\ ps ) <-> ( ph /\ ps ) ) ) |
16 |
11 14 15
|
3imtr4i |
|- ( ( [ a <> b ] ph /\ [ a <> b ] ps ) -> [ a <> b ] ( ph /\ ps ) ) |