Description: Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005)
Ref | Expression | ||
---|---|---|---|
Assertion | suble | |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lesubadd | |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) ) |
|
2 | lesubadd2 | |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( ( A - C ) <_ B <-> A <_ ( C + B ) ) ) |
|
3 | 2 | 3com23 | |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - C ) <_ B <-> A <_ ( C + B ) ) ) |
4 | 1 3 | bitr4d | |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) ) |