Description: Restricted quantifier version of Theorem 19.3 of Margaris p. 89. We don't need the nonempty class condition of r19.3rzv when there is an outer quantifier. (Contributed by NM, 25-Oct-2012)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | rr19.3v | |- ( A. x e. A A. y e. A ph <-> A. x e. A ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biidd | |- ( y = x -> ( ph <-> ph ) ) |
|
| 2 | 1 | rspcv | |- ( x e. A -> ( A. y e. A ph -> ph ) ) |
| 3 | 2 | ralimia | |- ( A. x e. A A. y e. A ph -> A. x e. A ph ) |
| 4 | ax-1 | |- ( ph -> ( y e. A -> ph ) ) |
|
| 5 | 4 | ralrimiv | |- ( ph -> A. y e. A ph ) |
| 6 | 5 | ralimi | |- ( A. x e. A ph -> A. x e. A A. y e. A ph ) |
| 7 | 3 6 | impbii | |- ( A. x e. A A. y e. A ph <-> A. x e. A ph ) |