Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018) (Proof shortened by Wolf Lammen, 29-Jun-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | ralanid | |- ( A. x e. A ( x e. A /\ ph ) <-> A. x e. A ph ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar | |- ( x e. A -> ( ph <-> ( x e. A /\ ph ) ) ) |
|
2 | 1 | bicomd | |- ( x e. A -> ( ( x e. A /\ ph ) <-> ph ) ) |
3 | 2 | ralbiia | |- ( A. x e. A ( x e. A /\ ph ) <-> A. x e. A ph ) |