Theorem elinintab

Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020)

	Ref	Expression
Assertion	elinintab	|- ( A e. ( B i^i |^| { x | ph } ) <-> ( A e. B /\ A. x ( ph -> A e. x ) ) )

Proof

Step	Hyp	Ref	Expression
1		elin	|- ( A e. ( B i^i |^| { x | ph } ) <-> ( A e. B /\ A e. |^| { x | ph } ) )
2		elintabg	|- ( A e. B -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) )
3	2	pm5.32i	|- ( ( A e. B /\ A e. |^| { x | ph } ) <-> ( A e. B /\ A. x ( ph -> A e. x ) ) )
4	1 3	bitri	|- ( A e. ( B i^i |^| { x | ph } ) <-> ( A e. B /\ A. x ( ph -> A e. x ) ) )