Step |
Hyp |
Ref |
Expression |
1 |
|
elpreq.1 |
|- ( ph -> X e. { A , B } ) |
2 |
|
elpreq.2 |
|- ( ph -> Y e. { A , B } ) |
3 |
|
elpreq.3 |
|- ( ph -> ( X = A <-> Y = A ) ) |
4 |
|
simpr |
|- ( ( ph /\ X = A ) -> X = A ) |
5 |
3
|
biimpa |
|- ( ( ph /\ X = A ) -> Y = A ) |
6 |
4 5
|
eqtr4d |
|- ( ( ph /\ X = A ) -> X = Y ) |
7 |
|
elpri |
|- ( X e. { A , B } -> ( X = A \/ X = B ) ) |
8 |
1 7
|
syl |
|- ( ph -> ( X = A \/ X = B ) ) |
9 |
8
|
orcanai |
|- ( ( ph /\ -. X = A ) -> X = B ) |
10 |
|
simpl |
|- ( ( ph /\ -. X = A ) -> ph ) |
11 |
3
|
notbid |
|- ( ph -> ( -. X = A <-> -. Y = A ) ) |
12 |
11
|
biimpa |
|- ( ( ph /\ -. X = A ) -> -. Y = A ) |
13 |
|
elpri |
|- ( Y e. { A , B } -> ( Y = A \/ Y = B ) ) |
14 |
|
pm2.53 |
|- ( ( Y = A \/ Y = B ) -> ( -. Y = A -> Y = B ) ) |
15 |
2 13 14
|
3syl |
|- ( ph -> ( -. Y = A -> Y = B ) ) |
16 |
10 12 15
|
sylc |
|- ( ( ph /\ -. X = A ) -> Y = B ) |
17 |
9 16
|
eqtr4d |
|- ( ( ph /\ -. X = A ) -> X = Y ) |
18 |
6 17
|
pm2.61dan |
|- ( ph -> X = Y ) |