Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994)
Ref | Expression | ||
---|---|---|---|
Assertion | freq2 | |- ( A = B -> ( R Fr A <-> R Fr B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 | |- ( A = B -> B C_ A ) |
|
2 | frss | |- ( B C_ A -> ( R Fr A -> R Fr B ) ) |
|
3 | 1 2 | syl | |- ( A = B -> ( R Fr A -> R Fr B ) ) |
4 | eqimss | |- ( A = B -> A C_ B ) |
|
5 | frss | |- ( A C_ B -> ( R Fr B -> R Fr A ) ) |
|
6 | 4 5 | syl | |- ( A = B -> ( R Fr B -> R Fr A ) ) |
7 | 3 6 | impbid | |- ( A = B -> ( R Fr A <-> R Fr B ) ) |