Theorem ralin

Description: Restricted universal quantification over intersection. (Contributed by Peter Mazsa, 8-Sep-2023)

	Ref	Expression
Assertion	ralin	|- ( A. x e. ( A i^i B ) ph <-> A. x e. A ( x e. B -> ph ) )

Proof

Step	Hyp	Ref	Expression
1		elin	|- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2	1	imbi1i	|- ( ( x e. ( A i^i B ) -> ph ) <-> ( ( x e. A /\ x e. B ) -> ph ) )
3		impexp	|- ( ( ( x e. A /\ x e. B ) -> ph ) <-> ( x e. A -> ( x e. B -> ph ) ) )
4	2 3	bitri	|- ( ( x e. ( A i^i B ) -> ph ) <-> ( x e. A -> ( x e. B -> ph ) ) )
5	4	ralbii2	|- ( A. x e. ( A i^i B ) ph <-> A. x e. A ( x e. B -> ph ) )