Theorem elintd

Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021)

	Ref	Expression
Hypotheses	elintd.1	|- F/ x ph
	elintd.2	|- ( ph -> A e. V )
	elintd.3	|- ( ( ph /\ x e. B ) -> A e. x )
Assertion	elintd	|- ( ph -> A e. |^| B )

Proof

Step	Hyp	Ref	Expression
1		elintd.1	|- F/ x ph
2		elintd.2	|- ( ph -> A e. V )
3		elintd.3	|- ( ( ph /\ x e. B ) -> A e. x )
4	3	ex	|- ( ph -> ( x e. B -> A e. x ) )
5	1 4	ralrimi	|- ( ph -> A. x e. B A e. x )
6		elintg	|- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )
7	2 6	syl	|- ( ph -> ( A e. |^| B <-> A. x e. B A e. x ) )
8	5 7	mpbird	|- ( ph -> A e. |^| B )