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 )