Description: Obsolete version of ralss as of 14-Oct-2025. (Contributed by Stefan O'Rear, 3-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ralssOLD | |- ( A C_ B -> ( A. x e. A ph <-> A. x e. B ( x e. A -> ph ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssel | |- ( A C_ B -> ( x e. A -> x e. B ) ) |
|
| 2 | 1 | pm4.71rd | |- ( A C_ B -> ( x e. A <-> ( x e. B /\ x e. A ) ) ) |
| 3 | 2 | imbi1d | |- ( A C_ B -> ( ( x e. A -> ph ) <-> ( ( x e. B /\ x e. A ) -> ph ) ) ) |
| 4 | impexp | |- ( ( ( x e. B /\ x e. A ) -> ph ) <-> ( x e. B -> ( x e. A -> ph ) ) ) |
|
| 5 | 3 4 | bitrdi | |- ( A C_ B -> ( ( x e. A -> ph ) <-> ( x e. B -> ( x e. A -> ph ) ) ) ) |
| 6 | 5 | ralbidv2 | |- ( A C_ B -> ( A. x e. A ph <-> A. x e. B ( x e. A -> ph ) ) ) |