Theorem rexin

Description: Restricted existential quantification over intersection. (Contributed by Peter Mazsa, 17-Dec-2018)

	Ref	Expression
Assertion	rexin	|- ( E. x e. ( A i^i B ) ph <-> E. 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	anbi1i	|- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ x e. B ) /\ ph ) )
3		anass	|- ( ( ( 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	rexbii2	|- ( E. x e. ( A i^i B ) ph <-> E. x e. A ( x e. B /\ ph ) )