Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997)
Ref | Expression | ||
---|---|---|---|
Assertion | poeq2 | |- ( A = B -> ( R Po A <-> R Po B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 | |- ( A = B -> B C_ A ) |
|
2 | poss | |- ( B C_ A -> ( R Po A -> R Po B ) ) |
|
3 | 1 2 | syl | |- ( A = B -> ( R Po A -> R Po B ) ) |
4 | eqimss | |- ( A = B -> A C_ B ) |
|
5 | poss | |- ( A C_ B -> ( R Po B -> R Po A ) ) |
|
6 | 4 5 | syl | |- ( A = B -> ( R Po B -> R Po A ) ) |
7 | 3 6 | impbid | |- ( A = B -> ( R Po A <-> R Po B ) ) |