Metamath Proof Explorer


Theorem ralanid

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 )

Proof

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 )