Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | relbrtpos | |- ( Rel F -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reltpos | |- Rel tpos F |
|
2 | 1 | a1i | |- ( Rel F -> Rel tpos F ) |
3 | brrelex2 | |- ( ( Rel tpos F /\ <. A , B >. tpos F C ) -> C e. _V ) |
|
4 | 2 3 | sylan | |- ( ( Rel F /\ <. A , B >. tpos F C ) -> C e. _V ) |
5 | brrelex2 | |- ( ( Rel F /\ <. B , A >. F C ) -> C e. _V ) |
|
6 | brtpos | |- ( C e. _V -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) |
|
7 | 4 5 6 | pm5.21nd | |- ( Rel F -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) |