Step |
Hyp |
Ref |
Expression |
1 |
|
subcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC ) |
2 |
|
subcl |
|- ( ( C e. CC /\ D e. CC ) -> ( C - D ) e. CC ) |
3 |
|
neg11 |
|- ( ( ( A - B ) e. CC /\ ( C - D ) e. CC ) -> ( -u ( A - B ) = -u ( C - D ) <-> ( A - B ) = ( C - D ) ) ) |
4 |
1 2 3
|
syl2an |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( -u ( A - B ) = -u ( C - D ) <-> ( A - B ) = ( C - D ) ) ) |
5 |
|
negsubdi2 |
|- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) ) |
6 |
|
negsubdi2 |
|- ( ( C e. CC /\ D e. CC ) -> -u ( C - D ) = ( D - C ) ) |
7 |
5 6
|
eqeqan12d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( -u ( A - B ) = -u ( C - D ) <-> ( B - A ) = ( D - C ) ) ) |
8 |
4 7
|
bitr3d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) = ( C - D ) <-> ( B - A ) = ( D - C ) ) ) |